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ELEMENTS OF ALGEBEA: 



ON 



THE BASIS OF M. BOURDOIf: 



EMBRACING 



^STURM^'S AND HORNER'S THEOREMS, 



AKD 



PRACTICAL EXAMPLES. 



BY CHAELES DAYIES, LL.D. 

AUTHOR OF ARITHMETIC, ELEMENTARY ALGEBRA, ELE^IENTART GEOHETRT, PRACTICAl 

GEOMETRY, ELEMENTS OF SURVEYING, ELEMENTS OF DESCRIPTIVE AND 

ANALYTICAL GEOMETRY, ELEMENTS OF DIFFERENTIAI. 

AND INTEGRAL CALCULUS, AND A TREATISE 

ON SHADES, SHADOWS AND PEK- 

SPECTIVE. 



N'EW YOEK: 
A. S. BARNES & COMPANY, 51, 53 & 55 JOHN STREET. 

BOLD BY BOOKSELLEKS, GENERALLY, TTIROUGIIOUT THE UNITED STATES. 

1866. 









Babies' primacy ^ritfjmetic and 2raI)Ie=3Soolt— Designed for Beginners; 
containing the elementary tables of Addition, Subtraction, Multiplication, 
Division, and Denominate Numbers ; with a large number of easy and prac- 
tical questions, both mental a,nd written. 

133b!PS' jFtrst 2-CSSons in ^ritftmctir — Combining the Oral Method with tlie 
Method of Teaching the Combinations of Figures by Sight. 

Gabies* ^-ntfUertual 0nt!)ntetic— An Analj-sis of the Science of Numbers, with 
especial reference to Mental Training and Development. 

iBabics' Nalu School i^vitf)ntctic — ^Analytical and Practical. 

B-n? to Babies' 'Mi'm Sdjcol ^uitljmctic. 

Sables' ©[rainmar of Sltitt)nTCtic — An Analysis of the Language of Niimbers 
and the Science of Figures. 

jDabics' Xcc) S^liiibersitu ^ritt)nictic — Embracing the Science of Numbers, and 
their Applications accordmg to the most Improved Methods of Analysis and 
Cancellation. 

K.C2> to Babies' !Metu ?3nibcrsity ^ritljmctic. 

Sables' UlCincutary .^Irjcbra— Embracing the First Principles of the Science. 

Itrj) to Dabtcs' Hlminitari) ^iQctra. 

Sab'^s' 5Hlcmentav2 ©ffcmetry a>cd ^riQonomrtrp — With Applications in 
Mensuration. 

Babies' 33'>-"Ctieal IHatJcmatics—Vrith Drawing and Mensuration applied to 
the Mechanic Arts. 

Babies* 5=liiibcrsit» ^Igefiva— Embracing a Logical Development of the 
Science, with graded examples. 

Babies' 333Uvtiou's .^13 cbr a— Including Sturm's and Horner's Theorems, 
and practical examples. 

Babus' acgentire's (Scomctri? anti 2ri-fcjonometr»—Eevised and adapted to 
the course of Mathematic.il Instruction in the United States. 

IDabfrs* Hlnnents of Sin begins a>-d "Xabfsatfou— Containing descriptions 
of the Instruments and necessary Tables. 

Babfes' SltinbJifcal CJcomctr^i — Embracing the Equations of the Point, the 
Straight Line, the Conic Sections, and Surfaces of the first and second order. 

Babies' Biffevcntial antd Jxutesval (Ealculus. 

Babies' Beseviptibe Geometry — With its application to Spherical Trigonome- 
try, Spherical Projections, and Warped Surfaces. 

Babies' S])a^cs, .S^aootos, and ^9a-speettbe. 

Babies' JLz^-z anti (HtiUtn of iHatI)rmaties~With the best methods of In- 
struction Explained and Illustrated. 

Babies' auU i3refe's <Hatl)eniatieaI B'rtfonari? anT) Cfelcpetifa of iUatt)r* 
mat cat Seiencc — Comprising Definitions of all tlie terms employed in 
Mathematics— an Analysis of each Branch, and of the whole, as foi-miag a 
single Science. 

EiTTEEKD according to Act of Congress, in tli? year one thousnnd eight hunc're.i and fiflv- 
one, by CnAnLES Daviks, in the Clerk's Ofiiee of tLe District foun of the United States 
for the Southern District of 2sew York. 



■VTiLLiAM Dexyse, Steeeotypee axd Ele -TrvOiYPER, 1S3 William Street, Xew York. 






PREFACE 



The Treatise on Algebra, by M. Bourdon, is a work 
of singular excellence and merit. In France, it liaa 
long been one of the standard Text books. Shortly after 
its first publication, it passed through several editions, 
and has formed the basis of every subsequent work on 
the subject of Algebra, both in Euroj)e and in this country. 

The original work is, however, a full and complete 
treatise on the subject of Algebra, the later editions 
containing about eight hundred pages octavo. The time 
which is given to the study of Algebra, in this country, 
even in those seminaries where the course of mathe- 
matics is the fullest, is too short to accomplish so volu- 
minous a work, and hence it has been found necessary 
either to modify it essentially, or to abandon it alto- 
gether. 

In the following work, the original Treatise of Bourdon 
has been regarded only as a model. The order of ar- 
rangement, in many parts, has been changed ; new rules 
and new methods have been introduced : the modifica- 
tions indicated by its use, for twenty years, as a text book 



op., / 

4 PREFACE. 

in the Military Academy have been freely made, for 
the ]3"i^^pose of giving to the -^vork a more practical 
character, and bringing it into closer harmony with the 
trains of thought and improved systems of instruction 
which prevail in that institution. 

But the work, in its present form, is greatly indebted 
to the labors of William G. Peck, A. M., U. S. Topo- 
graphical Engineers, and Assistant Professor of Mathe- 
matics in the Military Academy. 

Many of the new definitions, new rules and improved 
methods of illustration, are his. His experience as a 
teacher of mathematics has enabled him to bestow upon 
the work much valuable labor which will be found to 
bear the marks of profound study and the freshness of 
daily instruction. 

FlEHKTLL LAXDEffQ, J 

May, 1858. f 



oJri^. -' ^^M^...^y 



CONTENTS. 



[APTER I. 

DEFINITIONS AND PRELIMINARY REMARKS. 

ALGI3BA— Definitions — Explanation of the Algebraic Signs ,.. 1 — 28 

Similar Terms — Reduction of Similar Terms 28 — 30 

Theorems — Problems — Definition of — Problem 80 — 31 

CHAPTER II. 

ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION, 

Addition— Rule 81 — 3* 

Subtraction — Rule — Remark 35 — ^1 

Multiplication — Rule for Monomials and Signs 41 — 45 

Rule for Polynomials - 45 — 46 

Remarks — Theorems Proved 46 — 49 

Division of Monomials — Rule 49 — 53 

Signification of the Symbol a? 53 — 55 

Division of Polynomials — Rule 55 — 58 

Remarks on the division of Polynomials 58 — 59 

Of Factoring Pylynomials 59—60 

When m is entire, a"" — 6"* is divisible by a — h 60 — 62 

CHAPTER III. 

ALGEBRAIC FRACTIONS. 

Definition — Entire Quantity — Mixed Quantity , . . 62 — 68 

Reduction of Fractions 68 — 69 

To Reduce a Fraction to its Simplest Form 68 — L 

To Reduce a Mixed Quantity to a Fraction 68 — II. 

To Reduce a Fraction to an entire or Llixed Quantity 68 — III. 

To Reduce Fractions to a Common Denominator 68 — IV 

To Add Fractions 08— V. 

To Subtract Fractions 68— VL 



6 CONTENTS. 

AR-nCLM 

To Multiply Fractions 68— VIL 

To Divide Fractions 68— VIIL 

Results from adding to both Terms of a Fraction 70 — 11 

Symbols 0, oo and J 71— '^3 

CHAPTER IV. 

EQUATIONS OF THE FIRST DEGREE. 

Definition of an Equation — Different Kinds — Properties of Equations 72 — 71 

Solution of Equations 77 — 78 

Transformation of Equations — First and Second 78 — 80 

Resolution of Equations of the First Degree — Rule 81 

Problems involving Equations of the First Degree 81 

Equations with two or more Unknown Quantities 82 — 83 

Elimination — By Addition — By Subtraction — By Comparison 83 — 88 

Problems giving rise to Simultaneous Equations. .. .„ Page 96 

Indeterminate Equations and Indeterminate Problems 88 — 89 

Interpretation of jS'egative Results 89 — 91 

Discussion of Problems 91 — 92 

Inequalities 92 — 93 

CHAPTER V. 

EXTRACTION OF THE SQUARE ROOT OF NUMBERS. OF ALGEBRAIC QUAN- 
TITIES. TRANSFORMATION OF RADICALS OF THE SECOND DEGREE. 

Extraction of the Square Root of Numbers 93 — 96 

Extraction of the Square Root of Fractions 96 — 100 

Extraction of ths Square Root of Algebraic Quantities 100 — 104 

Of Monomials 100—101 

Of Polynomials 101—104 

Radicals of the Second Degi-ee 104 — 106 

Addition and Subtraction— Of Radicals 106—107 

Multiplication, Division, and Transformation 107 — 110 

CHAPTER VI. 

EQUATIONS OF THE SECOND DEGREE. 

Equations of the Second Degree .110 — 112 

incomplete Equations — & lution of 112 — 1 14 

Solution of Complete Equations of the Second Degree 114 — 115 

Dit^cussion of Equations of the Second Degree 1 15 — 117 

Of the Four Forms 1 17—121 

Problem of the Lights 121—122 

Of Trinomial Equations 122—126 

Extraction of the Square Root of the Binomial a ±-,/6 12.5 — 126 

Equations with two or more Uiiknown Quantities..... 3 26 — 128 



CONTENTS. 7 

CHAPTER VII. 

FORMATION OF POWERS, BINOMIAL THEOREM, EXTRACTION OF ROOTS 
OF ANT DEGREE WHATEVER. OF RADICALS. 

Formation of Powers, 128— 13C 

llieory of Permutations and Combinations 130 — 136 

Binomial Theorem 136—141 

Extraction of the Cube Roots of Numbers 141 — 142 

To Extract the w'* Root of a Whole Number 142—144 

Extraction of Roots by Approximation 144 — 145 

Extraction of the n'^ root of Fractions 145 — 146 

Cube Root of Decimal Fractions. . 146—14'? 

Extraction of Roots of Algebraic Quantities 147 — 148 

Of Polynomials 148—150 

Transformation of Radicals 150 — 16b 

Addition and Subtraction of RadicaU. 155 — 156 

Multiplication of Radicals , 156 — 151 

Division of Radicals 157 — 158 

Formation of Powers of Radicals 158 — 159 

Extraction of Roots 159 — 160 

Different Roots of the Same Power 160—162 

Modifications of the Rules for Radicals 162 — 164 

Theory of Fractional and Negative Exponents 164 — 171 

CHAPTER VIII. 

OF SERIES. ARITHMETICAL PROGRESSION. GEOMETRICAL PROPORTION 

AND PROGRESSION. RECURRING SERIES. BINOMIAL FORMULA. 

SUMMATION OF SERIES. PILING OF SHOT AND SHELLS. 

Series Defined 171—178 

Arithmetical Progression — Defined 172 — 173 

Expression for the General Term 174 — 176 

Sum of any two Terms 175 — I7d 

Sum of all the Terms 176—177 

Formulas and Examples 177 — 181 

Ratio and Geometrical Proportion 181 — 186 

Geometrical Progression — Defined 186 — 187 

Expression for any Terra 187 — 188 

Sum of n Terms — Formulas and Examplps 188 — 193 

Lideterminate Co-efficients 193 — 199 

Recurring Series 199 — 202 

General Demonstration of Binomial Tlieorem 202 — 204 

Applications of the Binomial Formula 204 — 208 

Summation of Series 208—209 

Metliod of Diffei mces 209—210 

Piling of Balls 210— 21« 



8 CONTENTS. 

CHAPTER IX. * 

CONIINUED FRACTIONS. EXPONENTIAL QUANTITIESv- —LOGARITHMS. 

Continued Fractions 215 — 224 

Exponential Quantities 224 — 227 

Theory of Logaritlims 227 — 229 

General Properties of Logarithms , 229 — 286 

Logarithmic Series — Modulus , 236 — 241 

Ti'ansformation of Series , 241 — 242 

Of Interpolation ; 242—243 

Of Interest .243—244 

CHAPTER X. 

GENERAL THEORY OF EQUATIONS. 

General Properties of Equations ,. 244 — 251 

Composition of Equations 251 — 252 

Of the Greatest common Divisor 252 — 262 

Transformation of Equations 262 — 264 

Formation of Derived Polynomials 264 — 266 

Properties of Derived Polynomials 266 — 267 

Equal Roots 267—270 

Elimination 270—276 

CHAPTER XI. 

SOLUTION OF NUMERICAL EQUATIONS. STURm's THEOREM. CARDAn's 

RULE. HORNEr's METHOD. 

General Principles. .1 275 — 277 

First Prmciple 277 — 279 

Second Principle 279—280 

Third Principle 280—281 

Limits of Real Roots 281—284 

Ordinary Limits of Positive Roots 284 — 285 

Smallest Limit in Entire Numbers 285 — 286 

Superior Limit of Negative Roots — Inferior Limit of Positive and 

Negative Roots 286—287 

Consequences , .287 — 293 

Descai'tes' Rule 293 — 295 

Commensurable Roots of Numerical Equations. , . , 295 — 298 

Sturm's Theorem , 298—308 

Cardan's Rule 308—309 

Preliminaries to Horner's Method .309 — 310 

Multiplication by Detached Co-efficients - 310 — 311 

Division by Detached Co-efficients , 311 — 312 

Synthetical Division 312—313 

Method of Transformation 313—314 

Homei's Metliod 314 



INTRODUCTIOJf 



Quantity is a general term applioa'ble to everything whicli> 
can be increased or diminished, and measured. There are two 
kinds of quantity; 

1st. Abstract quantity, or quantity, the conception of which 
does not involve the idea of matter ; and, 

2dly. Concrete quantity, which embraces every thing that is 
material. 

Mathematics is the science of quantity ; that is, the science 
which treats of the measurement of quantities, and of their 
relations to each other. It is divided into two parts : 

1st. The Pure Mathematics, embracing the principles of the 
science and all explanations of the processes by which these 
principles are derived from the abstract quantities, Number 
and Space ; and, 

2d. The Mixed Mathematics, embracing the applications of 
these principles to all investigations involving the laws of 
matter, to the discussion of all questions of a practical nature, 
and to the solution of all problems, whether they relate to 
abstract, or concrete quantity.* 

*Davies' Logic and Utility of Mathematics. Book IL 



10 INTRODUCTION. 

There are three operations of the mind which are imme 
diately concerned in the investigations of mathematical science : 
Ist. Apprehension; 2d. Judgment; 3d. Reasoning. 

1st. Apprehension is the notion, or conception of an idea 
iu the mind, analogous to the perception by the senses. 

2d. Judgment is the comparing together, in the mind, two 
of the ideas which are the objects of Apprehension, and pro 
uouncing that they agree or disagree with each other. Judg 
ment, therefore, is either affirmative or negative. 

Sd. Reasoning is the act of proceeding from one judgment 
to another, or of deducing unknown trmhs from principles al- 
ready known. Language affords the signs by which these opeia- 
tions of the mind are expressed and communicated. An appre 
kensioji, expressed in language, is called a term; a judgment, 
expressed in language, is called a proposition; and a p^ote-is 
of reasoning, expressed in language, is called a demon^ira- 
tlon* 

The reasoning processes, in Logic, are conducted usually by 
means of words, and in all complicated cases, can take place 
in no other way. The words employed are signs of ideas^ 
and are also one of the principal instruments or helps of 
thought; and any imperfection in the instrument, or in the 
mode of using it, will destroy all ground of confidence in the 
result. So, in the science of mathematics, the meaning of the 
terms employed are accurately defined, while the language 
arising from the use of the symbols, in each branch, has a 
definite and precise signification. 



* Whately's Logic, — of tie operations of the mind and senses. 



INTRODUCTION. 11 

In the science of numbers, the ten characters, called figures, 
are the alphabet of the arithmetical language ; the combinations 
of these characters constitute the pure language of arithmetic; 
and the principles of numbers which are unfolded by means 
of this, ir connection with our common language, constitute 
tlie science. 

It Geometry, the signs which are employed to indicate the 
boundaries and forms of portions of space, are simply the 
straight line and the curve; and these, in connection with our 
common language, make up the language of Geometry : a 
science which treats of space, by comparing portions of it 
with each other, for the purpose of pointing, out their proper 
ties and mutual relations. 

Analysis is a general term embracing that entire portion of 
mathematical science in which the quantities considered are 
represented by letters of the alphabet, and the operations to 
be performed on them are indicated by signs. 

Algebra, which is a branch of Analysis, is also a species 
of universal arithmetic, in which letters and signs are employed 
to abridge and generalize all processes involving numbers. It 
is divided into two parts, corresponding to the science and 
art of Arithmetic : 

1st. That which has for its object the investigation of the 
properties of numbers, embracing all the processes of reasoning, 
by which new properties are inferred from known ones ; and, 

2d. The solution of all problems or questions involving the 
detemaination of certain numbers which are unknown, from 
their connection with certain others which are known or given. 



12 INTRODUCTION. 

Li arithmet/c, all quantity is regarded as consisting of parts, 
which can be numbered exactly or approximatively, and in 
this respect, possesses all the properties of numbers. Proposi- 
tions, therefore, concerning numbers, have this remarkable pecu 
liarity, that they are propositions concerning all quantities 
whatever. Algebra extends the generalization still further. A 
number is a collection of things of the same kind, without refer- 
ence to the nature of the thing, and is generally expressed by 
figures. Algebraic symbols may stand for all numbers, or for all 
quantities which numbers represent, or even for quantities which 
cannot be exactly expressed numerically. 

In Geometry, each geometrical figure stands for a class ; 
and when we have demonstrated a property of a figure, that 
property is considered proved for every figure of the class. In 
Algebra, all numbers, all lines, all surfaces, all solids, may be 
denoted by a single symbol, a or x. Hence, the conclusions 
deduced by means of those symbols are true of all things what- 
ever, and not like those of number and Geometry, true only 
for particular classes of things. The symbols of Algebra, there- 
fore, should not excise in our minds ideas of particular things. 
The wi'itten characters, a, 6, c, c?, a:, y, 0, serve as the 
representatives of things in general, whether abstract or con- 
crete, whether known or unknown, whether finite or infinite. 

In the various uses which we make of these symbols, aiid 
the processes of reasoning carried on by means of them, the 
mind insensibly comes to regard them as things^ and not as 
mere signs ; and we constantly predicate ^f them the properties 
of things in general, without pausing to inquire what kind of 



INTBODUCTION. 13 

thing is implied. All this we are at liberty to do, since the 
symbols being the representatives of quantity in general, there 
is no necessity of keeping the idea of quantity continually alive 
in the mind; and the processes of thought may, without dan- 
ger, be allowed to rest on the symbols themselves, and there- 
fore, become to that extent, merely mechanical. But when we 
look back and see on what the reasoning is based, and how 
the processes have been conducted, we shall find that every 
step was taken on the supposition that we were actually 
dealing with things^ and not with symbols; and that without 
this understanding of the language, the whole system is without 
signification, and fails.* 

The quantities which are the subjects of the algebraic analysis 
may be divided into two classes ; those which are known or 
given, and those which are unknown or sought. The known 
are uniformly represented by the first letters of the alphabet, 
a, 6, c, c?, &c. ; and the unknown by the final letters, a:, y, 
g, V, &;c. 

Five operations, only, can be performed upcn a quantity 
that will give results differing from the quantity itself: viz. 
1st. To add a quantit- to it; 
2d. To subtract a quantity from it; 
3d. To multiply it by a quantity; 
4th. To divide it; 
5th, To extract a root of it. 

Five signs only, are employed to denote these operations. 
They are too well known to be repeated here. These, with 

• Davies' Logic and Utility of Mathematics. § 218. 



14 INTRODUCTION. 

the signs of equality and inequality, together with the letters of 
the alphabet, are the elements of the algebraic language. 

The interpretation of the language of Algebra is the first 
thing to which the attention of a pupil should be directed ; 
and he should be drilled in the meaning and import of the 
symbols, until their significations and uses arl as familiar as 
the sounds of the letters of the alphabet. 

All the apprehensions, or elementary ideas, are conveyed to 
the mind by means of definitions and arbitrary signs ; and 
every judgment is the result of a comparison of such impressions. 
Hence, the connection between the symbols and the ideas which 
►hey stand for, should be so close and intimate, that the one 
<5hall always suggest the other ; and thus, the processes of 
Algebra become chains of thought, in which each link tulfils the 
double ofllce of a distinct and connecting propog tion. 



/r. 



ELEMENTS OF ALGEBRA. 



CHAPTER I. 

DEFINITIONS AND PRELIMINARY REMARKS. 

1. Quantity is anything which can be increased or dimin- 
ished, and measured. 

2. Mathematics is the science which treats of the measurement 
and relations of quantities. 

3. Algebra is a branch of mathematics, in which the quantities 
considered are represented by letters, and the operations to be 
performed upon them are hidicated by signs. The letters and 
signs are called symbols. 

4. In algebra two kinds of quantities are considered: 

1st. Known quantities^ or those whose values are known or 
given. These are represented by the leading letters of the alplia/- 
bet, as, «, 5, c, &c. 

2d, Unknown quantities^ or those whose values are not given. 
They are denoted by the final letters of the alphabet, as, 
x, y, z, &c. 

T^etters employed to represent quantities are sometimes written 
with one or more dashes, as, a\ b", c'", x\ y", &c., and are 
read, a jn-ime, b second, c third, x prime, y second^ &c. 

5. The sign 4-, is called plus, and when placed between two 
quantities, indiaitcs that the one on the right is to be added to 
the cue on the loft. Thus, a -f 6 is read a plus b, and indicates 



16 ELEMENTS OF ALGEBRA. [CHAP. I, 

that the quantity represented by h is to be added to the quan- 
tity represented by a. 

6. The sign — , is called minus, and when placed between two 
quantities, indicates that the one on the right is to be subtracted 
from the one on the left. Thus, c — d is read c minus d, and 
indicates that the quantity represented by d is to bo subtracted 
from the quantity represented by c. 

The sign -}-, is sometimes called the positive sign, and the 
quantity before which it is placed is said to be positive. 

The sign — , is called the negative sign, and quantities affected 
by it are said to be negative. 

7. The sign X , is called the sign of multiplication, and when 
placed between two quantities, indicates that the one on the left 
is to be multiplied by the one on the right. Thus, a X b, indi 
cates that a is to be multiplied by b. The multiplication of 
quantities may also be indicated by placing a simple point 
between them, as a.b, which is read a multiplied by 6. 

The multiplication of quantities, which are represented by 
letters, is generally indicated by simply writing the letters one 
lifter another, without interposing any sign. Thus, 
ab is the same as a x b, or a.b; 
and abc, the same as a X b X c, or a.b.c. 

It is plain that the notation last explained cannot be employed 
when the quantities are represented by figures. For, if it were 
required to indicate that 5 was to be multiplied by 6, we 
could not write 5 6, without confounding the product with th« 
TJ umber 56. 

The result of a multiplication is called the product, and each 
of the quantities employed, is called a factor. In the product 
of several letters, each single letter is called a literal factor. 
Thus, in the product ab there are two literal factors a and b ; in 
the product bed there are three, b, c and d. 

8. The sign -'-, is called the sign of division, and when placed 
between two quantiti^es, indicates that the one on the left is to be 
divided by the one on the right. Thus, a~ b indicates that a is to 



CHAP. I.] DEFINITIONS AND KEMARKS. 17 

be divided by h. The same operation may be indicated by wi'iting 
b under a, and drawing a line between them, as — ; or by writing 
It on tho right of a, and drawing a Ime between them, as a\b, 

9. The sign =, is called the sign of equality^ and indicates that 
tno two quantities between which it is placed are equal to each 
other. Thus, a — 6 =: c + c?, indicates that a diminished b;y 6 is 
equal to c increased by d. 

10. The sign >, is called the sign of inequality^ and is used to 
indicate that one quantity is greater or less than another. 

Thus, a > 6 is read, a greater than h ) tuid a <^h is read, a less 
than h ; that is, the opening of the sign is turned toward the greater 
quantity. 

11. The sign <^ is sometimes employed to indicate the difference 
)f two quantities when it is not known which is the greater. 

Thus, a '^ b, indicates the difference between a and 6, without 
•howing which is to be subtracted from the other. 

12. The sign ex, is used to indicate that one quantity varies as 

to another. Thus a a -— , indicates that a varies as -— . 
b 

13. The signs : and : :, are called the signs of proportion ; the 
first is read, is to, and the second is read, as. Thus, 

a : b : : c : d, 
is read, a is to b, as c is to d. 

Ilie sign .'., is read hence, or consequently. 

14« If a quantity is taken several times, as 
a-{-a-\-a-{-a-{-a, 
it is generally written but once, and a number is then placed- 
before it, to show how many times it is taken. Thus, 

a + a + a-f-^ + a may be written 6a. 
The i;umber 5 is called the co-efficient of a, and denotes lliat a is 
taken 5 times. 

Hence, a co-efficient is a number prefixed to a quantity, denoting 
the number of times which the quantity is taken. 

2 



18 ELEMENTS OF ALGEBRA. LCHAP. L 

When no co-efficient is written, the co -efficient 1 is always under- 
stood; thus, a is the same as la. 

15. If a quantity is taken several times as a factor, the pr.>duct 
may be expressed by writmg the quantity once, and placing a 
number to the right and above it, to show how many times It is 
taken as a factor. 

Tlius, ay^ay^ay^ay^a may be written a^. 

The number 5 is called an exponent, and indicates that a is 
taken 5 times as a factor. 

Hence, an exponent is a number written to the right and above 
a quantity, to show how many times it is taken as a factor. \i 
no exponent is \mtten, the exponent 1 is understood. Tnus, a is 
the same as a^. 

16. If a quantity be taken any number of times as a factor, the 
resulting product is called a power of that quantity : the exponeni 
denotes the degree of the power. For example, 

oi> zi^a is the first power of a, 
a? z=.a y a is the second power, or square of a, 
a^ z=: a X ci X a is the third power, or cube of a, 
a* = aXaXaXa is the fourth power of a, 
a^i=zayayayaya is the fifth power of a, 
m which the exponents of the powers are, 1, 2, 3, 4 and 5 ; and 
the powers themselves, are the results of the multiplications. It 
should be observed that \h^ exponent of a power ys, always greater 
by one than the number of multiplications. The exponent of a 
power of a quantity is sometimes, for the sake of J)revity, called 
the exponent of the quantity. 

17. As an example of the use of the exponent in algebra, let 
it be required to express that a number a is to be multiplied 
tliree times by itself; that this product is then to be multiplied 
three times by 6, and this new product twice by c ; we should 
write 

aXaXaxax6x^'X6xcXcrr: a^i^c^. 
If it were further required to take this result a certaiu number 
of times, say seven, we should simply write 7a*6V 



CHAP. I.] DEFINITIONS AND REMARKS. 10 

I8i A root of a quantity, is a quantity wliicli being taken a 
certain number of times, as a fi.ctor, will produce the given 
quantity. 

The sign -/^is called the radical sign, and when placed o\er 
a quantity, indicates that its root is to be extracted. Thus, 
^l/H, or simply Va denotes the square i oot of a. ' 
^/a denotes the cube root of a. 
*/a denotes the fourth root of a. 
The number placed over the radical sign is called the imkx 
of the root. Thus, 2 is the index of the square root, 3 of tljo 
cube root, 4 of the fourth root, &c. 

19. Tlie reciprocal of a quantity, is 1 divided by that quantity. 
Thus, 

— is the reciprocal of a; 

and J- is the reciprocal of a -f ft. 

a-f-o 

20i Every quantity written in algebraic language, that is, by 
the aid of letters and signs, is called an algebraic quantity^ or tlie 
algebraic expression of a quantity. Thus, 

(is the algebraic expression of three times the 

t quantity denoted by a ; 

( is the algebraic expression of five times the 

[ square of a ; 

( is the algebraic expression of seven times the 

( product of the cube of a and the square of 6; 

_ _* ( is the algebraic expression of the difference 
6a — 56 -^ . 

( between three times a and five times 6; 

is the algebraic expression of IvAcq the square 

^ , « , . ..n . oi a, diminished by three times the produet 

of a and 6, augmented by four times the 



I 



square of b. 

21 1 A single algebraic expression, not connected with any other 
by the sign <jf addition or subtraction, is called a monomial, 01 
simply, a term. 



20 ELEMEISTS OF ALGEBRA. [CHAP. I 

Thus, oa, 5a2, Ta^i^^ are monomials, or single terms. 

An algebraic expression composed of t^vo or more terms cod^ 
nected by the sign + or — , is called a polynomial. 

For example, oa — bb and Sa^ — och + 46^, are poh nomials. 

A polynomial of two terms, is called a binomial; and one of 
three terms, a trinomial. 

22* The numerical value of an algebraic expression, is the num 
ber obtained bj givmg a particular value to each letter which 
enters it, and performing the operations indicated. This numer- 
ical value will depend on the particular values attributed to the 
letters, and will generally vary ^dth tliem. 

For example, the numerical value of 2a'^, will be 54 if we make 
a = 3; for, 3^ = 3 X 3 X 3 = 27, and 2 X 27 = 54. 

Tlie numerical \alue of the same expression is 250 when we 
make a — 5; for, 5^ = 5x5x5— 125, and 2 X 125 = 250. 

We say that the numerical value of an algebraic expression 
generally varies with the values of the letters which enter it; it 
does not, however, always do so. Thus, in the expression a ■ 6, 
so long as a and h are increased or diminished by the same 
number, the value of the expression will not be changed. 

For example, make a = l and 6 = 4: there results a — b — Z. 

Now, make a m 7 + 5 =: 12, and 6 = 4 + 5 = 9, and there 
results, as before, a — 6 = 1 2 — 9 = 3. 

23. Of the dilferent terms which compose a polynomial, some 
are preceded by the sign +, and others by the sign — . The 
former are c^'.lled additive terms, the latter, subtractive terms. 

When the first term of a polynomial is plus, th^ sign is gene- 
rally omitted ; and when no sign is written before a term, it is 
always understood to have the sign -f-. 

24. The numerical value of a polynomial is not affected by 
changmg the order of its terms, provided the signs of all the 
terms remain unchanged. For example, the polynomial 

4a3 _ 3a26 + 5ac2 = 5ac2 — 3(^6 + 4a3 = — 3a-6 + bac"^ + Aa\ 

25. Each literal factor v^hich enters a term, is called a dimen- 
Bion of the term ; and the degree of a term is indicated by tlie 
n'jmber of these factors or dimensions. Thus, 



CHAP. I.] DEFINITIONS AND REMARKS. 2l 

3a is a term of one dimension, or of the first degree. 

5ab is a term of two dimensions, or of the second degree. 

la^bc^ = laaahcc is of six dimensions, or of the sixth degree. 

In general, the degree of a term is determined hy taking the sum 
of the exponents of tJie letters which enter it. For example, the 
term Sa-bcd^ is of the seventh degree, since the sum of the expo- 
nents, 

2+1 + 1+3, is equar to 7. 

26« A polynomial is said to be homogeneous, when all of its 
terms are of the same degree. The polynomial 

3a — 26 + c is homogeneous and of the first degree. 

— Aah + b"^ is homogeneous and of the second degree. 

ba^c — 4c^ + 2c^c? is homogeneous and of the third degree. 

8a3 — 4ai + c is not homogeneous. 

27o A vinculum , parenthesis ( ), brackets [], j }, or 

bar I, may be used to indicate that all the quantities which they 
connect are to be considered together. Thus, 

a-\- b -\- c X X, {a -\-b -\- c) X X, [a + 5 + c] X ar, or {a -\- b -{- c} x, 
indicate that the trinomial a + 6 + c is to be multiplied by x. 

Wiien the parenthesis or brackets are used, the sign of mul- 
tiplication may be omitted: as, {a-\- b ■\- c)x. The bar is used 
in some cases, and differs from the vinculum in being placed 
vertically, as + a x. 

+ c 

28. Terms which contain the same letters affected with equal 

exponents are said to be similar. Thus, in the polynomial, 

lab + Zab — 4.aW + ^aW, 

the tenns lab and 3a6, are similar, and so also are the terms 

— 4a^6' and Sa^i^, the letters in each being the same, and thft 

same letters being affected with equal exponents. But in tho 

binomial 

QaV) + lab\ 

the terms are not similar; for, although they contain the same 
letters, yet the same letters are not affected with equal expo- 
nents. 



22 ELEMENTS OF ALGEBRA. [CHAP. U 

29. When a polynomial o^ntains similar '.erms, it may be 
reduced to a simpler form by forming a single term from each 
set of similar terms. It is said to be in its simplest form, when 
it contains the fewest terms to which it can be reduced. 

li we take the polynomial 

2a^c'^ — 4aHc^ + 6a^c^ — Sa^bc^ + lla^lc^, 
we know, from the definition of a co-efficient, that the literal 
part a^c'^ is to be taken additively, 2 rf 6 + 11, or 19 times j 
and subtractively, 4 -\- 8, or 12 times. 
Hence, the given polynomial reduces to 

19a^c^ — 12a35c2 = 7a^c\ 
It may happen that the co-efficient of the subtractive term, ob- 
tained as above, will exceed that of the additive term. In that 
case, subtract the positive co-efficient from the negative, prefix the 
fninus sign to the remainder, and then annex the literal part. 
In the polynomial 

3a26 + 2a2^--5a26-3a26, 
we have, + oa'^b — ho?b 

+ ^o?b — 3a26 



+ 5a26 — 8a26 

But, — Sa26 = — ba^b — Za-b : hence 

5a26 - 8a25 = ba^b - ba% - Za% = - Za'^b. 

In like manner we may reduce the similar terms of any poly- 
nomial. Hence, for the reduction of a polynomial containing 
sets of similar terms, to its simplest form, we have the following 

PtULE. 

I. Add together the co-efficients of all the additive terms of each set^ 
and annex to their sum the literal part : form a single subtractive 
term in the y2me manner. 

II. Then, subtract the less co-effi.cient from the greater, and to the 
remainder prefix the sign of the greater co-efficient, and annex the 
literal vart. 



CHAP. I.] REDUCTION OF POLYNOMIALS. 28 

EXAMPLES. 

1. Ecduce the polynomial Aa^ — Sa^b — 9a^b + llaP'b lo its 
simplest form. A71S, — 2a'^b. 

2. Reduce the polynomial labc^ — abc^ — labc"^ — Saic^ + Ga6c' 
to its simplest form. Ans. — Sabc^. 

3. Reduce the polynomial 9cb^ — Sac^ + 15cb^ -f 8ca + 9ac'^ 

— 24ci3 to its simplest form. Ans. ac^ + 8ca. 

4. Reduce the polynomial Qac^ — 5a6^ + "^^c^ — Sai^ — ISac^ 
\- \Qab^ to its simplest form. Ans. \Oab\ 

5. Reduce the polynomial abc^ — ahc -\- ^ac^ — ^abc"^ -\- Qtahc 

— 8ac2 to its simplest form. Ans. — Sabc"^ -\- 5abc — Sac^. 

6. Reduce the polynomial ^a^^ — 7a^b + 5ab — 9a^b^ + 9a^ 
-f- Sab to its simplest form. Ans. — Qa^b"^ + 2a^b + 8a5. 

7. Reduce the polynomial Sacb^ — la^c%'^ — Q>a'^b^ — Zacb* 
4 Qa?c%^ — Qacb'^ + 4a''i^ + 2a*5^ to its simplest form. 

Ans. — ah%^ — Qacb'^. 

8. Reduce the polynomial — la^b^c^ 4- 9a^bc^ + Ga'^b'^c^ -\- a^b'^c^ 

— ha^bc^ — 6^c^ to its simplest form. Ans. 4a^bc^ — b^c^. 

9. Reduce the polynomial — lOa^ + Qa^^ + "^^^^ — Sa^^^ 
- 5a% -f oa^Z*^ to its simplest form. Ans. — 8a^ + 4:a'^b^. 

Remark. — It should be observed that the reduction affects only 
the co-efficients, and not the exponents. 

30. A TiiEOP,p-M is a general truth, which is made evident by a 
course of reasoning called a demonstration. 

A PROBLEM is a question proposed which requires a solution. 

31. We shall now illustrate the utility and brevity of algebraic 
language by solving the following 

PROBLEM. 

The sum of two members is 67, and their difference is 19 ; whai 
are the numbers ? 

Let us first indicate, by the aid of algebraic symbols, the 
relation which exists betwee-T the given and unknown numbers 
of the problem. 



24 ELEMENTS OF ALGEBRA. [CHAP, i. 

If the less of the two numbers were known, the greater could 

be found by adding to it the difference 11^ , or in other w(-rds, 

tlie less number, plus 19, is equal to the greater. 

If, then, we denote the less number by x, 

a; + 19 will denote the greater, 

and 2a; + 19 will denote the sum. 

But from the enunciation, this sum is to be equal to 67. Thero 

i'ore. 

2:r + 19 = 67. 

Now, if 2x augmented by 19, is equal to 67, 2.r alone is equal 

\a} 67 mmus 19, or 

2.r=z:67-19, 

or performing the subtraction, 

2.C = 48. 

Hence, x is equal to half of 48, that is 

48 ^. 
. = - = 24. 

The less number being 24, the greater is 
a: + 19 = 24 + 19 = 43. 
And, indeed, we have 

43 + 24 = 67, and 43 - 24 = 19. 

GENERAL SOLUTION". 

The sum of two numbers is a, and their difference is h Wliat 
are the two numhers ? 

Let X denote the less number ; 

Then will x -{- b denote the greater number. 

Now, from the conditions of the problem, 
x-\- X + h, or 2x -\-b 
'A ill be equal to the sum of the two numbers : herje, 

^x + b = a. 
Now, if 2.r + 5 is equal to a, 2x alone must be equal t# 
a —h and 

a — h _ a b 

"^ ^ "~2~ ~ 2" ~ y 



CHAP. I.J SOLUTION OF PEOBLEMS. 25 

If the value of x be increased by 6, we shall have the 
gi'eater number : that is, 

, a h - ah 

hence, x -\- h = -— -\- — — the greater number, and 

ir = — — z=. the less number. 

That is, the greater of two iiumbers is equal to half their sum 
increased by half their difference ; aitd the less is equal to half 
ilieir sum diminished by half their difference. 

As the form of these results is independent of any particular 
values attributed to the letters a and 6, the expressions are called 
formulas^ and may be regarded as comprehending the solution 
of all problems of the same kind, differing only in the numerical 
values of the given quantities. Hence, 

A formula is the algebraic expression of a general rule, or 
principle. 

To apply these formulas to the case in which the sum is 237 

«md difference 99, we have 

237 99 237 + 99 336 ,^^ 
the greater number — — [- — = — — — = 168 ; 

^ ^ , 237 99 237-99 138 
and the less =—-- — := = -^ '-^ 69 ; 

and these are the true numbers ; for, 

168 4- 69 =; 237 which is the given sum, 
Qud 168 -- 69 = 99 whicl: :s the given differencis^ 



CHAPTER n. 

ADDITION, SLBTt ACTION. MULTIPLICATION, £jtD DIVISION. 

ADDITI02T. 

31 • Addition, in algebra, is the operation of finding the sim- 
plest equivalent expression for the aggregate of two or more alge- 
braic quantities. Such equivalent expression is called their sum, 

32. If the quantities to be added are dissimilar, no reductions 
can be made among the terms. \Ve then write them one 
after the other, each with its proper sign, and the resulting 
polynomial will be the simplest expression for the sura. 

For example, let it be required to add together the mono- 
inials 

3a, 55 and 2c ; 
we coimect them by the sign of addition, 

3a + 56 + 2c, 
a result which cannot be reduced to a simpler form. 

33* If some of the quantities to be added have similar terms, 
we connect the quantities by the sign of addition as before, 
and then reduce the resulting polynomial to its simplest form, 
by the rule already given. This reduction will, in general, be 
more readily accomplished if we write down the quantities to 
be added, so that similar terms shall fall in the same column. 
Thus; 

Let it be required to find the sum of \ 

«K f-r 1 2a2_Sa5-h52 

the quantities, t 

Their sum, after reducing (Art. 29), is - ba^ — bah — 45^ 



GHAP II.] ADDITION. 27 

34. As operations similar to the above apply to all algebraic 
expressions, we deduce, for the addition of algebraic quantities, 
the following general 

RULE. 

I. Write down the quantities to he added, with their respective 
signs, so that the similar terms shall fall in the same column. 

II. Reduce the similar terms, and annex to the results tf'ose term* 
which cannot be reduced, giving to each term its respectivt sign. 

EXAMPLES. 

I. Add together the polynomials, 

Sa^-2b^-4ab, 6a^ — b"^ -]- 2ab and Sab — Sc^ — 2b\ 



The term Sa^ being similar to 5a^ 



3^2 _ 4^1 _ 2%^ 

5«2 4- 2ub — &2 

+ Sub - 252 



8a2 _|_ ab - 5^2 _ 3c2 



we write 8^2 for the result of the re- 
duction of these two terms, at the same ^ 
time slightly crossing them as in the 
terms of the example. 

Passing then to the term — 4ab, which is similar to the two 
terms -f- 2ab and + Sab, the three reduce to + «^, which is 
placed after Sa^, and the terms crossed like the fn-st term. 
Passing then to the terms involving b^, we find their sum to be 
— 5^2^ after which we write — 3c2. 

The marks are drawn across the terms, that none of them 
may be overlooked and omitted. 



Sum 



feura 



(2). 




(3). 


7a; + 3a6+ 2c 




16a2/>2 4_ bc-2abc 


— Sx- Sab — 5c 




— 4a252 _ Qf^c ^ Q^ic 


5x — 9ab— 9c 




- 9^27,2 + 6c + abc 


. 9x — 9ab — 12c 




i3a252 _ 7^c _|_ 5^^^ 


(4). 




(5). 


a+ ab— cd-\- f 


6ab + cd+ d 


Sa + hab — Qcd - 


■ / 


Sab-{- bcd — y 


— 5a - (Sab + Qcd - 


-V 


— 4a6-f Qcd-^x 


— a -h a.h -^ cd-\-4f 


- hah — 12cc^ + y 


- 2a -1- ai + - 


i¥ 


-{-x + d 



28 ELEMENTS OF ALGEBEA. [CHAP. IL 

6. Add together Za + 6, 3a + 35, - 9a — 76, Qa + 95 and 
8a + 35 + 8c. Ans. Ua + 9b -\-Sc. 

7. Add together 3a.r -f 3ac +/, — 9aa: + 7a + (/, -\- Qaz f 3a« 
+ 3/, Sax + 13ac + 9/ and — 14/+ Sax. 

Afis. Uax + 19 jc — /+ 7a + d. 

8. Add together the polynomials, Sa^c + 5a5, 7a^c — 3a5 + 3a« 
Sa^c — Qab + 9ac and — Sa^c + a5 — 12ac. Ans. la^c — 3a5. 

9. Add the polynomials, l^a^x^b — 12a3c5, 5a^x^b + iDahb 

— ^Oax, — 2a'^x^b — IZahb and — 18a2.c35 — 12a3c5 + 9 a^. 

Ans. 4:a^x^b — 22a^cb — ax. 

10. Add together 3a + 5 + c, 5a + 25 + Sac, a -\- c -\- ac and 

— 3a — 9ac — 85. Ans. Ga — 55 + 2c — 5ac. , 

11. Add together 5a25 + 6c^ + 95c2, 7c.z; — 8a25 and — 15c.c 
-95c2 + 2a25. Ans. -a^ — 2cx. 

12. Add together Sax + 5a5 + SaWc"^, — l^ax + Ga^ + lOoi 
and lOa^*— 15a5 - Ga^iV. Ans. — Sa'^hH'^ + Ga2. 

13. What is the sum of ^\aWc — 2^iahc —Wd^y and l^a^h^c 
+ 9a5c1 Ans. hlaWc — \Sabc —14:ay. 

14. What is the sum of 18a5c — 9a5 + 6c^ — 3c + 9ax and 
9a5c + 3c — 9ax ? Ans. 27abc — 9ab + 6c\ 

15. What is the sum of Sabc -j- b^a — 2cx — Gxi/ and 7c2 

— xy — 1353a ] A^ns. Sabc — 12Pa + ocx — 7xy. 
IG. What is the sum of 9a2c — 14a5y + ISa^^^ and — a^c 

-80252? Ans. 8a2c - 14a5y + 7a252. 

17. What is the sum of 17a552 + 9a35 — 3a2, _ 14a-52 + 7a' 

— 9a3, — 15a35 + '7a^b^- — a^ and 14a35 — 19a35 ? 

Ans. . 

18. What is the sum of Sax-^ — 9ax^ — I7axy, + 9a.r2 + 18aa^ 
+ S4.axy and 'Ta^b + Sax^ — 7a.r2 + 45ca: 1 Ans. . 

19. Add together 3a2 + 5a252c2 — 9a3.r, 7a2 — 8a2i2c2 _ lOa-'^.r 
and 10a5 + 16a252c2 + 19a3.r. A?is. lOa.2 + 13a252c2 + 10a5. 

20. Add together 7a25 — 8a5c — 852c — 9c3 + cd% Sc.bc — 5aH 
+ 3c3 - 452c + cd^, and 4a25 - Sc^ + 952c — Sd^. 

A?is. Qa'^b + 5a5c - Sl'^^ -Uc^ + 2ccr' - 3fP. 






CHAP. II.J SUBTRACTION. • 29 

21. Add together - 18a^ -^ 2ab^ -i- (ja^\ -8ab* + 7a^ -^a^^ 
aiid — 5a^6 + Gab^ ^- UaW. Ans. — Ua^ + 12aW. 

22. What is the sum of Sa^^c — ICnt^x — dax^d, + GaWc 
•- r)aa:\Z + .7a*a; and + IGax^d — a^x — Sa^h 1 

Ans. a%^c + ax^d. 

23. What is the sum of the following terms : viz., 8a^ — 10a*& 
- IQfcW 4- 4a2i3 _ i2a*5 + 15a362 _^ ^^^.W — 6a6* - 16a363 
4 20a?63^32rt&*-865'? 

^71*. 8a5 - 22a^b - lla^U^ + 48a263 + 26a64 _ 8&5. 

•■ O 



SUBTRACTION. ""^ 

35. Subtraction, in algebra, is the operation for finding the 
simplest expression for the difference between two algebraic 
qiuantities. This difference is called the remainder. 

36. Let it be required to subtract 46 from 5a. Here, as 
the quantities are not similar, their difference can only be indi- 
cated, and we write * 

5a - 46. 

Again, let it be required to subtract Ao?b from la^b. These 
ierms being similar, one of them may be taken from the other 
and their true difference is expressed by 
^a?l - 4a36 = Za%. 

37* Generally, if from one polynomial we wish to subtract 
another, the operation may be indicated by enclosing the second 
in a parenthesis, prefixing the minus sign, and then writing it 
after the first. To deduce a rule for performing the operation 
thus mdicated, let us represent the sum of all the terms in the 
first polynomial by a. Let c represent the sum of all the ad- 
ditive terms in the other polynomial, and — d the sura of 
the subtractive terms ; then this polynomial will be represented 
by c — d. The operation may then be indicated thus, 

a — {c - d^\ 
where it is required to subtract from a the difference between 
c and d. 



30 ELEMENTS OF ALGEBRA. ^CH.VP. IL 

If, now, we diminish the quantity a by the quantity c, the 

result a — c will be too smal by the quantity d^ since c should 

have been diminished by d before taking it from a. Hence, 

to obtain the true remainder, we must increase the first result 

by d^ which gives the expression 

a — c -}- c?, 
and this is the true remainder. 

By comparing this remainder with the given polynomials, we 
see that we have changed the signs of all the terms of the quantity 
to be subtracted, and added the result to the other quantity. To 
facilitate the operation, similar quantities are written m the sikme 
column. 

Hence, for the subtraction of algebraic quantities, we have the 
following 

RULE. 

I. Write the quantity to he subtracted under that from which ti 
is to be taken, placing the similar terins^ if there are any, i^ the 
same column. 

II. Change the signs of all the terms of the quantity to be sub- 
tracted, or conceive them to be changed, and then add the result to 
the other quantity, 

EXAMPLES. 

(1). Ifl (!)• 

From - %ac — 5a6 4- c^ « o| Qac — bab -\- c^ 

Take - Zac -f oab — 7c ".|| — 3ac — Zab + 7c 

Remainder Zac — 8a6 + c2 + 7c. II | oac — 8a6 + c^ -f 7c. 

(2). " (3). 

From - 16a2 — 56c -f- 7ac \9abc — \Qax — 5a.ry 

Take • Ma^ + 55c + 8ac 17q6c + lax-l haxy 

Remamder 2a2 — 106c — ac 2abc — 2Sax + lOaxy 

(4). (5). 

From - 5a3 — 4a^b + Sb^c 4ab — cd -{- Sa^ 

Take - - 2a3 + 3a^6 - 86^c 5a5 — Acd + 3a^ -f 56^ 

, Remamder 1 a^ - 1 a'-h ^- Ub'^c — "^6^3^^ 0~ - 5^2. 



CHAP. II.] SUBTRACTION. 31 

6. From Sa^x — l^ahc + 7a2, take 9a^x — IZahc. 

Ans. — Qa^x -+- 7a'. 

7. From 51a262c — \Sahc — \4:a^y, take AAdWc — llahc 

— lA.a'^y. Ans. WaWc + 9abc. 

8. From 27a6c — 9a6 + ^c"^? take 9ahc -)- 3c — 9aa:. 

yl/is. 18a6c — 9a6 + 6c2 — 3c + 9a.r. 

9. From 8a6c — 12^3^* + bcx — 7ary, take 7ca; — xy — 136^a. 

^ns. Sa^c + Pa — 2cx — Qxy. 
10 From ^ah — \Aahy + la'^h'^, take 9a2c — l^aby + 15a262. 

Aris. — a?c — Sa^S^, 

11. From 9tt6a:2 — 13 + 20a63.r — A.hhx'^, take S^^co^^ + ^a^x^ 
^ 6 + 3a63a:. ^7i5. 17a6'^2' — Ih'^cx'^ — 7. 

12. From 5a* ^- 7tt3i2 ._ 3c5^2 4. 7^^^ take 3a4 - 3a2 - Ic^d^ 

— l^aW. Ans. 2a* + 8a362 + 4:C^d'^ + 7cZ + 3a2. 

13. From 51a262 _ 4Sa36 + 10a*, take 10a* - ^a^^h — Qa^^. 

Ans. 57a262 _ 40a36. 

14. From 2lx3y^ + 2bxhj^ + 68:ry* — 402/^, take Q>\x'^y^ 
H- 48.ry* - 40y^ Ans. 20ry* — 39.t2z/3 + ^Ix^y"^. 

15. From 53a:V _ l^xhj^ — 18:r*y — 56a;5, take — 15ar2y3 
-f l^xhf + 24a:*y. ^7i5. 35a;3y2 _ /^^x^y _ 50a?5. 

38* From what has preceded, we see that polynomials may be 
subjected to certain transformations. 

For exampla - . - . 6a2 — 3a6 + 2^2 — 26c, 
may be written - - - - 6a2 — (8a6 — 262 _|. 25c). 
In like manner - . . . 7a3 — 8a26 — 462c + ^)63, 
may be written .... 7a3 — (8^26 + 462c — 06-^) y 

or, again, la"^ — Sa'^b — [4.h^c — (jP). 

^Iso, 8a2 - 6a262 + 5a263, 

becomes 8a2 — (6a262 — 5a263). 

Also, - 9a2r3 — 8a* 4- 62 — c. 

may be written - - - - 9a2c3 — (8a* — 62 -f c) ; 
or, it may be wi'itten . - 9a2c3 -|- 6^ — (8a* + c). 

Tliese transformations consist in separating a polynomial inU) 
two parts, an'^i then connecting the parts by the minus sign. 



82 ELEMENTS OF ALGEBRA. ICHAP. IL 

It will be observed that the sign of each term is changed when 
the term is placed witliin the parenthesis. Hence, if we have 
one or more terms included within a parenthesis having the 
minus sign before it, the signs of all the terms must be changed 
when the parenthesis is omitted. 

Thus, 4a - {Qab - 3c - 2b), 

Is equal to 4a — 6a5 + 3c + 26. 

Also, Gab — {—4ac + Sd — 4ab), 

is equal to Gab -\- 4ac — Sd + 4ab. 

39. Eemark. — From what has been shown in addition and 
subtraction, we deduce the following principles. 

1st. In Algebra, the words add and sum do not always, as in 
arithmetic, convey the idea of augmentation. For, if to a we add 
— 6, the sum is expressed by a — b, and this is, properly speaking, 
the arithmetical difference between the number of units expressed 
by a, and the number of units expressed by b. Consequently, 
tliis result is actually less than a. 

To distinguish this sum from an arithmetical sum, it is called 
the algebraic sum. 

Thus, the polynomial, 2a^ — Sa^ + Sb\ 
is an algebraic sum, so long as it is considered as the result of 
the union of the monomials 

2a3, — 3a26, + Sb\ 
with their respective signs; but, in its proper acceptation, it is 
the arithmetical difference between the sum of the units con- 
tained in the additive terms, and the units contained in the 
subtractive term. 

It follows from this, that an algebraic sum may^ in the numer 
ical applications, be reduced to a negative expression. 

2d. Tlie words subtraction and difference, do not always convey 

the idea of diminution. For, the difference between -f- « and 

— b being 

a — ( — 6) = a + 6, 

is numerically greater than a. This result is an algebraic differ^ 
ence. 



CHAP. II.] MULTIPLICATION 83 

40» It frequently occurs in Algebra, that the algebraic sign -\- 
or — , which is written, is not the true sign of the teim before 
which it is placed. Thus, if it were required to subtract — h 
from a, we should write 

a — { — h) = a-\-b. 
Here the true sign of the second term of the binomial is plus, 
although its algebraic sign is — . This minus sign, operating 
upon the sign of 6, which is also negative, produces a plus sign 
for b in the result. The sign which results, after combining the 
algebraic sign with the sign of the quantity, is called the essen- 
tial sign of the term^ and is often different from the algebraic 
sign. 



MULTIPLICATION. 

41. Multiplication, in Algebra, is the operation of finding the 
product of two algebraic quantities. The quantity to be multi- 
plied is called the multiplicand ; the quantity by which it is 
multiplied is called the multiplier ; and both are called factors. 

42. Let us first consider the case in which both factors are 
monomials. 

Let it be required to multiply 7a^'^ by 4a'^b ; the operation 
may be indicated thus, 

KaW X 4a^, 
or by resolving both multiplicand and multiplier into their 
simple factors, 

laaabb X 4aab. 
Now, it has been shown in arithmetic, that the value of a 
product is not changed by changmg the order of »ts factors ; 
hence, we may write the product as follows: 

7 X 4:aaaaabbb, which is equivalent to 28a^b^. 
Comparing this result with the given factors, we see that the 
co-efficient in the product is equal to the product of the co-effi- 
cients of the multiplicand and multiplier ; and that the exj)oneut 
of each letter is equal to the sum of the exponents of that letter 
in both multiplicand and multiplier. 

3 



u 



ELEMENTS OF ALGEBRA. 



[CHAP. II. 



And since the same course of reasoning may be applied to 
any two monomials, we have, for the multiplication of mono 
iclals, the following 

RULE. 

J. M'Lltiph, the co-efficients together for a new co-efficient. 

11. Write after this co-efficient all the letters which enter into the 
multiplicand and multiplier, giving to each an exponent equal to 
the sum of its exponents in both factors. 

EXAMPLES. 

(1) - - 8a25c2 X 7a6cf2 ^ ^QaWc'^d'^. 
{2) • - 21aW-dc X 8abc^ = USa^b^c^d. 



Multiply- 
by . 



(3) 
Sa% 
2ba^ 



(4) 
. 12a^x 
- 12a:2y 



(5) 
6xi/z 

ai/z 



(6) 

a^xi 
2xy'^ 



a^xy 



Qaxy'^ 



Ans. ^Qa^W'c'-d. 

Ans. 60abcd^. 

Ans. la^b'^d\*. 



\4Aa^x'^y 

7. Multiply 8a^b\ by la%Hd. 

8. Multiply babd? by \2cd^. 

9. Multiply 7a^bd^c^ by abdc. 

43. We will now proceed to the multiplication of polynomiaia. 
In order, to explain the most general case, we will suppose the 
multiplicand and multiplier each to contain additive and sub- 
tractive terms. 

Let a represent the sum of all the additive terms of the multi- 
plicand, and — b the sum of the subtractive terms ; c the sura 
of the additive terms of the multiplier, and — d the sum of 
the subtractive terms. The multiplicand will then be represented 
by a — 6 and the multiplier, by c — d. 

We will now show how the multiplication expressed by 
[a — b) X {c — d) can be effected. 

The required, product is equal to a — 6 
taken as many times as there are units 
in c — d. Let us first multiply by c ; 
that is, take a — b as many times as 
there are units in c. We begin by writ- 
ing ac, which is too great by b taken 



a 


-b 




c 


-d 




ac 


-be 






— ad-\-bd 


ac 


^bc- 


• od^bd. 



CHAP. II.] MULTIPLICATION. 36 

c times ; for it is only the difference between a and 6, that is 
first to be multiplied by c. Hence, ac — be is the product of 
a — h by c. 

But the true product is a — 6 taken c — d times : hence, the 
last product is too great hj a — b taken d times ; that is, by 
ad — bd^ which must, therefore, be subtracted. Suotracting this 
from the first product (Art. 37), we have 

(a — b) X {c — d) =z ac — be — ad -{- bd : 

If we suppose a and c each equal to 0, the product will re 
duce to -f- bd. 

44. By considering the product of a — b by c — c? , we may 
deduce the following rule for signs, in multiplication. 

When two terms of the multiplicand and multiplier are affected 
with the same sign^ their product will be affected with the sign -f, 
and when they are affected with contrary signs, their product will 
be effected with the sign — . 

We say, in algebraic language, that + multiplied by + 
or — multiplied by — , gives + ; — multiplied by +, or -\- mul 
tiplied by — , gives — . But since mere signs cannot be multi- 
plied together, this last enunciation does not, in itself, express a 
distinct idea, and should only be considered as an abbreviation 
of the preceding. 

This is not the only case in which algebraists, for the sake of 
brevity, employ expressions in a technical sense in order to se- 
cure the advantage of fixing the rules in the memory. 

45. We have, then, for the multiplication of polynomials, the 
following 

RULE. 

Multiply all the terms of the multiplicand by each term of the 
mnltiplier in succession, affecting the product of any two terms with 
tlvfi sign plus, when their signs are alike, and with the sign minus, 
when their signs are unlike. TJien redu:e the polynomial result 
to its simj^lest form. 



^^ ELEMENTS OF ALGEBRA. [CHAP. II, 

EXAMPLES. 

1. Multiply 3a2 + 4a5 -I- 62 

by 2a + 56 

6a^ + SaH + 2ab'^ 

+ 15a^6 + 20ab^ 4- 56» 
Product 6a^ -f 23a^6 + 22a62 + 56^ 

(2). (3). 

a:^ 4- y^ ^^ + ^?/^ + '7aa; 

X — 1/ ax -{- 5ax 

x^ + xy"^ ax^ -\- ax^y^ + 7a^x^ 

— x'^y — y^ 4- 5aa;^ + 5ax^y^ -f- 35a^a;* 

a?3 4- ^y^ — ^^y — y^ 6«^^ 4- Qax^y^ 4- 42a2a:2. 

4. Multiply a;2 4- Sao; -{- a^ by x -\- a. 

A71S. x^ 4- 3a^2 _j_ 3q^2^ _j_ 0^3^ 

5. Multiply a;2 4- ?/2 ijy ^ _^ y^ 

^715. a;3 4- ^y^ 4- ^V + y^* 

6. Multiply 3a62 4- Qah'^ by 3a62 4- Sa^c"^. 

Ans. 9fb^ 4- 27a362c2 4- 18a*c*. 

7. Multiply 4a;2 — 2y by 2y. Ans. Sx^y — 4y\ 

8. Multiply 2x 4- 4y by 2^: — 4y. Ans. 4^2 _ 10^2^ 

9. Multiply x^ 4- ^^y 4- ^y"^ 4- y^ by a; — y. Ans. , 

10. Multiply a;2 4- ary 4- y^ by x^ — ^y -\- y^. 

Ans. x^ + x^y^ + y*. 

In order to bring together the similar terms, in the product o 
two polynomials, we arrange the terms of each polynomial witn 
reference to a particular letter ; that is, we arrange them so tha 
the exponents of that letter shall go on diminishing from left 
to right. 

11 Multiply 4a3- 5a26 - 8a62 4-263 

by 2a2 - 3a6 ~ 46^ 

Sa^ — lOa'^b — }6aW-^ 4.a?b^ 

— 12a*6 4- 15a'^62 + 24a263 - 6a6* 

— 16g^62 4- 20q2^,3 _^ 32^54 ■_ §51 

8a5 — 22a*6 — 17^362 _|_ 48^253 + 26a6* — 86*. 



^zii^^'^'*-^^^ 



CHAP. ILJ ^MfL^IPLIC^ON. 37 

After having ariangcd the polynomials, wifch reference to the 
letter a, multiply each term of the first, Ly the term 2a^ of the 
second; this gives the polynomial Sa^ — lOa^b — l^a^^ -\- \.a^l?^ 
ui which the signs of the terms are the same as in the multi- 
plicand. Passing then to the term — Zah of the multiplier, mul- 
tiply each term of the multiplicand by it, and as it is affected 
with the sign — , affect each product with a sign contrary to 
that of the corresponding term in the multiplicand ; this gives 

- 12a*5 + \^aW -f "l^aW — ^ah^. 
Multiplying the multiplicand by — 4^^^ gives 
- \^aW + 20^2^3 _|_ 32a^»4 _ 865. 

The product is then reduced, and we finally obtain, for the most 
simple expression of the product, 

8a5 - 22a46 - MaW + ^^a'h'^ + 26a5* - 865. 

12. Multiply 2a2 — Zax + ^x^ by Sa^ — ^ax — 2x\ 

Ans. 10a* - 27a^x + Ma^x^- — ISax^ - 8x^, 

13. Multiply Sx^ - 2ya; + 5 by x^ -f 2x?/ — 3. 

Ans. Sx^ + 4x^j/ — 4:X^ — A.x^y'^ -f \^xy — 15. 

14. Multiply Zx^ + 2a;2y2 4. 3^2 ]t)y 2x^ — 3^2y2 4. 5^3, 
6a;6 _ 5a;V _ e^j^y* + 6a;3?/2 4. 15:c3y» 

9^V* + 10a;22/5 + ISy^. 

15. Multiply Saar — 6a5 — c by 2ax -\- ah -{■ a. 

Ans. I6a^x^ — 4a^x — QaW + 6acx — lahc — c\ 

16. Multiply 3a2 — 562 _{_ 3^2 \^j ^,2 _ ^,3, 

Ans. 3a* — 5a262 + 3a2c2 — 3a263 + 56^ — 36-V. 

17. Multiply 3a2 — ^hd + cf 
by — 5a2 + 46^; — 8c/. 

Product — 1 5a* + Zlci?hd — 29ahf — 20b^cP + 446cc//— siy*. 

18. Multiply 4a-^62 — 5a262c + 8a26c2 - 3a2c3 — 7a6c3 
by 2a62 — 4a6c - 26c2 + c^. 

8a*6* — 10a36*c + 28a3^,3c2 _ S4a^^c^ 
l^oduct -j — 4a263c3 — 16a*63c + 12a36c* + 7a262c* 
4-14a26c5 +14a62c5.— 3a2c6 — 7a6c6. 



Ans. 



j 6a;' 



38 ELEMENTS OF ALGEBRA. [CHAP. IL 

46* REMARKS ON THE MULTIPLICATION OF POLYNOMIALS. 

Isi. If both multiplicand and multiplier are homogeneous^ the 
product will be homogeneous, and the degree of any term of the 
product will be indicated by the sum of the numbers which indicate 
the degrees of its two factors. 

Thus, in example 18th, each term of the multiplicand is of 
the 5th degree, and each term of the multiplier of the 3d de- 
gree : hence, each term of the product is of the 8th degree. 
This remark serves to discover any errors in the addition of 
the exponents. 

2c:?. If no two terms of the product are similar, there will be no 
reduction amongst them ; and the number of terms in the product 
will then be equal to the number of terms in the multiplicand^ multi 
plied by the member of terms in the multiplier. 

This is evident, since each term of the multiplier will produce 
as many terms as there are terms in the multiplicand. Thus, in 
example 16th, there are three terms in the multiplicand and two 
in the multiplier : hence, the number of terms in the p 'o«^uct is 
equal to 3x2 = 6. 

Zd. Among the terms of the product there are always two which 
cannot he reduced with any others. 

For, let us consider the product with reference to any letter 
common to the multiplicand and multip'..*ier : Then the irreduci- 
ble terms are, 

1st. The term produced by the multiplication of the two terms 
of the multiplicand and multiplier which contain the highest 
power of this letter ; and 

2d. The terni produced by the multiplication of the two terms 
which contain the lowest power of this letter. 

For, these two partial products will contain this letter, to a 
higher and to a lower power than either of the other partial pro 
ducts, and consequently, they cannot be similar to any of them. 
This remark, the truth of which is deduced from the law of 
tb»^ exponents, will be very useful in division. 



CHAP. II.] 


MULTIPLICATION. 




EXAMPLE. 


Multiply - 
by - - 


. . 5a<b^ + Sa'b - ab* - 2ab^ 
a26 - a62 


Product 





39 



If we examine the multiplicand and multiplier, with reference 
to a, we see that the product of 5a^b^ by a'^b, must be irre- 
ducible ; also, the product of — 2ab^ by ab"^. If we consider 
the letter &, we see that the product of — ab^ by — a^^, must 
be irreducible, also that of Sa^b by a^. 

47t The following formulas depending upon the rule for mul- 
tiplication, will be found useful in the practical operations of 
algebra. 

Let a and b represent any two quantities ; then a + b will 
represent their sum, and a — b their difference. 

I. We have {a + by = (a + b) X {a -\- b), ; '>! 
or performing the multiplication indicated, 

{a -f by = a2 -f- 2ab + b^ ; that is. 
The square of the sum of iivo quantities is equal to the square 
of the first, plus twice the product of the first by the second, plus 
tlie square of the second. 

To apply this formula to finding the square of the binomial 
5a2 + 8a25, 
we have (5a2 -f ^a^bf = 25a* + SOa^b -[- 64a^b\ 
Also, {6a^b + 9ab^Y = S(ja%^ + lOSa^M + Sla^K 

II. We have, (a - by = {a ~ b) X {a - b), 
or performing the multiplication indicated, 

{a - by =za^- 2ab + b^ ; that is, 
The square of the difference between two quantities is equal (0 
the square of the first, minus tiuice the product of the first bxj tht 
second, plus the square of the second. ^ "^ - 
To apply this to an example, we have 

(7a2Z,2 _ I2a63)2 = 49a4i4 - IGSa^Js ^ I4,4.a'^h^. 
Also, (4c363 7cV^)2 = lQa%^ - bQaWcH^ + A9cH\ 



40 ELEMENTS OF ALGEBPwA. [CHAP. IL 

III. We have {a -i- b) X {a -b) = a^ - b\ 
hy performing the multiplication ; that is, 

The sum of two quantities multiplied by their difference is equal 
io the difference of their squares. 

To apply this formula to an example, we have 

(8a3 + 7a52) x {Sa^ - 7ab^) = 64.a^ - 4Qa^¥. 

48. By considering the last three results, it is perceived 
tiiat their composition, or the manner in which they are formed 
from the multiplicand and multiplier, is entirely independent of 
any particular values that may be attributed to the letters a and 
&, which enter the two factors. 

The manner in which an algebraic product is formed from its 
two factors, is called the law of the product ; and this law re- 
mains always the same, whatever values may be attributed to 
the letters which enter into the two factors. 



DIVISION. 

49» Division, in algebra, is the operation for finding from two 
given quantities, a third quantity, which multiplied by the second 
shall produce the first. 

The first quantity is called the dividend^ the second^ the divisor^ 
and the third, or the quantity sought, the quotient. 

50. It was shown in multiplication that the product of two 
terms having the same sign, must have the sign +5 and that 
the product of two terms having unlike signs must have the 
sign — . Now, since the quotient must have such a sign that 
when multiplied by the divisor the product ^vill have the sign of 
the dividend, we nave the following rule for signs in division. 

If the dividend is -f and the divisor -r the quotient is -f- 
if the dividend is + and the divisor — the quotient is — 
if the dividend is — and the divisor + the quotient is ~ 
if the dividend is — and the divisor — the quotient is -(-. 

That is : The quotient of terms having like signs is plus, ana 
the quotient of terms having unlike signs is mims. 



CHAP. II.J DIYISION. 41 

51. Let us first consider the case in which both dividend and 
divisor are monomials. Take 

ooa^Jj^c^ to be divided by la^bc; 
The operation may be indicated thus, 

Now, since the quotient must be such a quantity as multiplied 
by the divisor will produce thr dividend, the co-efRcient of the 
quotient multiplied by 7 mus*. give 35 ; hence, it is 5. 

Again, the exponent of each better in tie quotient must be such 
that when added to the exponent of the same letter in the divisor, 
the sum will be the exponent of that letter in the dividend. 
Hence, the exponent of a in the quotient is 3, the exponent of 
6 is 1, that of c is 1, and the required quotient is ^a%c. 

Since we may reason in a similar manner upon any two 
monomials, we have for the division of monomials the following 

RULE. 

I. Divide the co-efficient of the dividend by the co-efficient of th€ 
divisor^ for a new co-efficient. 

II. Write after this co-efficient^ all the letters of the dividend 
and give to each an exponent equal to the excess of its expo 
nent in the dividend over that in the divisor. 

By this rule we find, 

A^aWc'^d , ,, , \hOa%\d'^ . o., , 

EXAMPLES. 

1. DIvid 16a;2 by 8.r. Ans. 2x. 

2. Divide loa'^xy^ by 3ay. Ans. 5axy^ 

3. Divide S4:aPx by 12^2^ Ans. 7abx. 

4. Divide — 96a*i»V by 12a^c. An-. —Qa'^bc'^. 

5. Divide \4:^a%^c'^d^ by — ZQ>a*b^c^d, Ans. — Aa^b^cd^. 

6. Divide — 256a^c^x^ by — IQa'^cx^. Ans. IQabcx, 

7. Divide — SOOa^b'^c^x^ by 30a4&V.r. Ans. — lOabcx. 
8 Divic^e — 400a%^c^x^ by 2oa%^c^x. A?is, -]66c2r*. 



42 ELEMENTS OF ALGEBRA. [CHAP. IL 

52<i It follows from the preceding rule that the exact division 
of monomials will be impossible : 

1st. When the coefficient of the dividend is not divisible by 
that of the divisor. 

2d. When the exponent of the same letter is greater in the 
divisor than in the dividend. 

This last exception includes, as we shall presently see, the 
case ill which the divisor has a letter which is not contained 
in the dividend. * 

When either of these cases occurs, the quotient remains un- 
der the form of a moiiomial fraction ; that is, a monomial 
expression, necessarily containing the algebraic sign of division. 
Such expressions may frequently be reduced. 

Take, for example, -g^- = -^. 

Here, an entire monomial cannot be obtained for a quotient; 
for, 12 is not divisible by 8, and moreover, the exponent of e 
is less in the dividend than in the divisor. But the expression 
can be reduced, by dividing the numerator and denominator by 
the factors 4, a^, 6, and r, which are common to both terms 
of the fraction. 

In general, to reduce a monomial fraction to its lowest terms: 
Sujypress all the factors common to loth numerator and denomi- 
nator. 

From this rule we find, 

4S>aWcd^ _ 4a^2 ^ ^lab^cH _ ZW^c 

36^6We ~ U^ ' ' 6a^bc*d^ ~ ~^M ' 

\2a%^c-' _ 3a6 ^ la% _ J._ 

^ ^^' IQa'b^c^ ~ 4^' ^^^ * Ia^W ~ 2ah ' 

In the last example, as all the factors of the dividend are 
found in the divisor, the numerator is reduced to 1 ; for, in fact, 
both terms of the fraction are divisible by the numerator. 

53. It often happens, that the exponents of certain letters, 

are the SAme in the divideiid and divisor. 

24a362 
tOT example, . - - . ^— , 



CHAP. II.J DIVISION. 43 

is a case iu which the letter h is affected with the same expo- 
nent in the dividend and divisor : hence, it will divide out, and 
will not appear in the quotient. 

But if it is desirable to preserve the trace of this letter in 
the quotient, we may apply to it the rule for exponents (Art. 
51), which gives 

62 

- = 62-2 ^ 50. 

Tlie symbol h^^ indicates that the letter b enters times as 
* factor in the quotient (Art. 16) ; or what is the same thing, 
that it does not enter it at all. Still, the notation shows that h 
was in the dividend and divisor with the same exponent, and 
has disappeared by division. 

In like manner, ^ ^, ' = ^a%'^c^ = 562. 

54t We will now show that the power of any quantity whose 

exponent is 0, is equal to 1. Let the quantity be represented 

by a, and let m denote any exponent whatever. 

a"* 
Then, — = a^~^ = a°, by the rule for division. 

But, —ml, since the numerator and denominator are equal : 

nence, a° = 1, since each is equal to 

We observe again, that the symbol a^ is only employed con- 
ventionally, to preserve in the calculation the trace of a letter 
which entered m the enunciation of a question, but which may 
disappear by division. 

55. In the second place, if the dividend is a polynomial and 
the divisor is a monomial, we divide each term of the dividend 
hy the divisor, and connect the quotients hy their resjjective signs. 

EXAMPLES. 

Divide 6a^x*y^ — I2a^xhj^ + 15a^xh/ by Sa^xhj^. 

Ajis. 2x^y* — 4ax]/^ -f 5a^x^y. 



44 ELEMENTS OF ALGEBRA. I CHAP. IL 

DIv'ide 12a*y6 — IQaY + 20a6y* - 2Sa''y^ by — 4.a^f. 

A71S. — 32/3 _J_ 4^y2 _ 5Qj2y _j_ 7^,3^ 

Divide ISa^Jc — 20acy2 -f 5cg?2 "by — 5^,5^^ 

42/2 ^^2 

^715. — 3a + -4^ -. 

ah 

56. In the third place, when both dividend and divisor are ' 
polynomials. As an example, let it be required to divide 

2Q,aW + 10a* — A&a^b + 24a63 by 4a6 - ha^ + 3^*2. 
Ill order that we may follow the steps of the operation more 
easily, we will arrange the quantities with reference to the letter a. 

Dividend. Divisor. 

10a* — 48a36 + 26a2Z*2 + 24ai3 | |— 5a2 + 4a5 + 3^2 

It follows from the definition of division and the rule for the 
multiplication of polynomials (Art. 45), that the di\"idend is the 
sum of the products arising from multiplying each term of 
the divisor by each term of the quotient sought. Hence 
if we could discover a term in the dividend which was derived, 
without reduction, from the multiplication of a term of the divi 
sor by a term of the quotient, then, by dividing this term o'' 
the dividend by that term of the divisor, we should obtain one 
term of the required quotient. 

Now, from the third remark of Art. 46, the term 10a*, con 
taining the highest power of the letter a, is derived, without 
reduction from the two terms of the divisor and quotient, con- 
taining the highest power of the same letter. Hence, by dividing 
the term 10a* by the term — 5a2, we shall have one term of 
the required quotient. 

Dividend. Divisor. 



10a* — 48a36 + 26a262 + 24a63 
f 10a* — 8a36 — 6«262 



-5a2 + 4a6 + 352 



— 2a2 4- ^ab 
40a^^6 + 32a2i2 ^ 24a63 Quotient. 

AOa^ + 32^252 4- 24a63. 



Smce the terms 10a* and — 5a2 are iffected with contrary 
signs, their quotient will have the sign — ; hence, 10a*, divided 
by — 5a2, gives — 2a2 for a ea-m of the required quotient. 



CBAP. II.] DIVISION. 45 

After having written this term under the divisor, multiply each 
term of the divisor by it, and subtract the product, 

10a* — Sa^b + Qa^% 
from the dividend. The remainder after the first operation is 

— 40a36 + ^2aW + 24aR 

This result is composed of the products of each term of the 
divisor, by all the terms of the quotient which remain to be 
determined. We may then consider it as a new dividend, and 
reason upon it as upon the proposed dividend. We will there- 
fore divide the term — 4:0a^b, which contains the highest power 
of a, by the term — 5a^ of the divisor. 

This gives + Sab 

for a new term of the quotient, which is written on the right 
of the first. Multiplying each term of the divisor by this term 
of the quotient, and writing the products underneath the second 
dividend, and making the subtraction, we find that nothing re- 
mains. Hence, 

— 2a2 + Sab or Sab — 2a^ 

is the required quotient, and if the divisor be multiplied by it, 
the product will be the given dividend. 

By considering the preceding reasoning, we see that, in each 
operation, we divide that term of the dividend which contains 
the highest power of one of the letters, by that term of the 
divisor containing the highest power of the same letter. Now, 
- we avoid the trouble of looking out these terms by arranging 
both polynomials with reference to a certain letter (Art. 45), 
which is then called the leading letter. 

Since a similar course of reasoning may be had upon any two 
polynomials, we have for the division of polynomials the following 

RULE. 

I. Arrange the dividend and divisor with reference to a certain 
letter, and then divide the first term on the left of the dividend by 
the first term on the left of the divisor, for the first term of the 
quotient ; multiply the divisor by this term and subtract the pro 
dud from the dividend. 



4:6 ELEMENTS OF ALGEBRA. [CHAP. II. 

a 

U. Then divide ike Jirst term of the remainder hy the first term 
of the divisor, for the second term of the quotient ; multiply the 
divisor hy this second term, and subtract the product from the 
result of the first operation. Continue the sa7ne operation until a 
remainder is found equal to 0, or till the first term of the remainder 
is not exactly divisible by the first term of the divisor. 

In the first case, (that is, when the remainder is 0,) the 
division is said to be exact. In the second case the exact divi- 
sion cannot be performed, and the quotient is expressed by 
writing the entire part obtained, and after it the remamder with 
its proper sign, divided by the divisor. 

SECOND EXAMPLE. 

Di^dde 21a;3y2 ^ ^bx^y^ + 6S:cy* — 40yS — 56a;5 — IS:^*^^ by 

— 40?/5 + 68ary* + 'H^x^y^ -|- 2lx'^y'^ — IQx^y — 56a;5| 15^2 _ ^xy—^x^ 

— 40y5 -[- 48:ry* + 64a;2y3 _ g^3 _j_ ^r^y2 _ Zx^y~^^^ 
1 St rem. 20.?;2/* — 39a;2y3 _^ ^\x^yi 

20.r?/* — 24a;2y3 _ ^^xhf 
2d rem. - — Xhx^y^ + 53a:3y2 _ \<^x^y 



3d. rem. .... Zhxhf — 42x'^y — 5Qx^ 

35a;3y2_42a:*y — 56a;5 

Final remainder - - - - - 0. 

57. Remark. — In performing the division, it is not necessary 
to bring down all the terms of the dividend to form the first 
remainder, but they may be brought do^vn in succession, as in 
the example. 

As it is important that beginners should render themselves 
familiar ^-ith algebraic operations, and acquire the habit of calcu- 
lating promptly, we will treat this last example in a different 
manner, at the same time, indicating the simplifications which 
should be introduced. These consist in subtracting each partial 
product from the dividend as soon as this product is formed. 



CHAP. II.] DIVISI03y>. 47 

— 40ys 4- 08xy« + 25.1-2?/? + 2lx^y^ — ISx^t/ — 56a;5| jSy^ _ 6.ry— 8a:« 

1st rem. 20.1^ — 39a;V4-21.rV ^8~yM^4^P^^^^3^^Vf^ 

2d rem. - ~15x'^y^ + 5Sx"-i/^ - I8x*y 

3d rem. .... 35.r^y2 _ 42.^.4^ _ S^.-^s 

Final remainder - - - 0, 

First, by dividing — 40y^ by 5?/^^ we obtain — Sy^ for the 
quotient. Multiplying 5?/2 by — Sy", we have — 40y^, or, by 
changing the sign, + 40y^, which cancels the first term of the 
dividend. 

In like manner, — Qxt/ x — Sy^ gives -|- 48.ry'^, or, changing 
the sign, — 48.'cy*, which reduced with + 08.^?/*, gives 20.Ty* for 
a remainder. Again, — Sx^ x — 8y^ gives +, and changing the 
sign, — 64x^y^, which reduced* with 25.^*2^3^ gives — Z9xhj^. 
Hence, the result of the first operatis^n is 20^y* — Sdx^y^, fol 
lowed by those terms of the dividend which have not been 
reduced with the products already obtained. For the second 
part of the operation, it is only necessary to bring down the 
next term of the dividend, to separate this new dividend from 
the primitive by a line, and to operate upon this new dividend in 
the same manner as we operated upon the primitive, and so on, 

THIRD EXAMPLE. 

Divide - - - 95a - 7Sa^ + 56c* — 25 - 59a3 by — 3a* 
4-5- Ua + '7a\ 

56a4 - 59a3 - 7Sa^ + 95a - 25 1 1 la^ - Sa^ - 11a 4- 5 



1st rem. — SSa^ 4- ISa^ -f 55a - 25 
2d remainder . - 0. 



8tt — 



GENERAL EXAMPLES. 

1 Divide 10a6 4 15ar by 5a. Ans. 2b + 3c. 

2 Divide 30aa; — 54.-1; by C)X. Ans. 5a — 9. 
S.Divide lO^'^y — 15y2 _ 5y by 5y. Ans. 2x'^ — ^1/ ~ 1. 
4. Divide 12a 4- 3aar — ISaa;'^ by 3a. Ans. 4 -\- x — (Sx^ 



18 ELEMENTS OF ALGEBRA. LCHAP. II 

5. Divide 6ax^ -\- 9a^x + a'^x^ by ax. Ans. Qx -\-^a-\- ax, 

6. Divide a^ -f- 2aa; -{■ x"^ by a -\- x. Ans. a -{- x, 

7. Divide a^ — Za^y -f Say^ _ y^ by a — y. 

Ans. a^ — 2ay + y^» 

8. Divide 24a26 — I'la^ch'^ — 6a6 by — Qah. 

Ans. — 4a + 2a2c6 + 1. 

9. Di\'ide 6^* — 96 by Zx — 6. ^ns. 2^^ _^ 4^2 ^ g^ _|_ le. 

10. Divide - - a^ — 5a*a; + lOa^^^ _ iq^^^s 4. 5^,^ _ ^ 
by a^ — 2a;r -{- x^. Ans. a^ — Sa^x + Saa;^ — a;^. 

11. Divide 48x-3 — 76a:K2 _ ^4^2^ _|_ i05a3 by 2x — Sa. 

Ans. 24^2 —2ax — S6a\ 

12. Divide y^ _ ^^^^2 ^ 3y2^4 _ ^e \yj yS _ 3^2^ _^ 3^,^2 _ ^^^ 

-4;is. 3/3 + Sy-a; + Sya:^ + x^, 

13. Divide 64a*66 _25a268 by Sa^3 -}- 5ab\ 

Ans. 8a263 _ ^abK 

14. Di\ade Ga^ + 23a25 + 22a52 + 563 ^y Sa^ + 4ab + b\ 

Ans. 2a -\- 56. 

1 5. Divide Qax^ + ^ax^y^ + 42a2.c2 by aa; + bax. 

Ans. x^ + xy^ + 7ca;. 

16. Divide - 15a* + ZHa'^bd - 29a'^cf--20b^d2 + 4Abcdf- Sc^P 
by 3a2 — 5bd + c/. Ans. — 5a2 + 46c? — 8f/. 

17. Divide x^ + x'^y^ + 2/* by x^ — xy -\- y"^. 

Ans. x"^ -\- xy -\- y^. 

18. Divide a;* — y* by a; — y. Ans. x"^ + ^^^y + ^y^ + 2/^ 

19. Divide Sa* — 8a262 + Sa^cZ -f 5i* _ 362c2 by a2 — b\ 

Ans. 3a2 - 562 _|_ 3^2, 

20. Divide Qx^ — 5x^y^ — 6x*y* + 6x^y^ + I5x^y^ — 9x^y* 
+■ 10a;2y5 -h 15y5 by 3a;3 + 2x^y^ + 8y2. 



CHAP, n.j DIVISION. 49 

REMARKS ON THE DIVISION OF POLYNOMIALS. 

68t The exact division of one polynomial by another is impossible: 

1st. When the first term of the arranged dividend or the first 
term of any of the remainders^ is not exactly divisible by the first 
term of the arranged divisor. 

It may be added wltli respect to polynomials that we win 
often discover by mere inspection that they are not divisible.. 
When the polynomials contain two or more letters, observe" the 
two terms of the dividend and divisor, which contain the highest 
powers of each of the letters. If these terms do not give an 
exact quotient, we may conclude tliat the cxuct division is im- 
possible. 

Take, for example, 

12a3 - 5^25 + 7a62 _ \W | |4a2 + 8fl6 + 3^A 

By considering only the letter a, the division would appear 
possible; but regarding the letter 5, the exact division is impos- 
sible, since —\\P is not divisible by Si^. 

2rf. When the divisor contains a letter which is not in the dividend.. 

For, it is impossible that a third quantity, multiplied by 
one which contains a certain letter, should give a product inde- 
pendent of that letter. 

Zd, A monomial is never divisible by a polynomial. 

rt>r, every polynomial multiplied by either a monomial or u 
poly:iomial gives a product containing at least two terms which 
are not susceptible of reduction. y" 

Aiih. If the letter^ with reference to which the dividend is ar- 
ranged, is not found in the divisor, the divisor is said to be inde- 
pendent of that letter ; and in that case, the exact division is 
impossible, 'Unless the divisor will divide separately the co-efficients 
of the dijferent powers of the leading letter. 

For example, if the dividend were 

36a* -f 96a2 + 126, 
arranged with reference to the letter a, and the div.'sor 36, the 
divisor would be independent of the letter a; and it is evident 

4 



50 ELEMENTS OF ALGEBRA. [CHAP. 



1 



that the exact division could not be performed unless the co- 
efficients of the different powers of a were exactly divisible by Zh, 
The exponents of the different powers of the leading letter 
in the quotient would then be the same as in the dividend. 

EXAMPLES. 

1. Divide i%a^x^ ^ZQaH^ — l2ax by Qx. 

Arts. Za^x — Ga^rc^ — 2a. 

2. Divide 25^46 — ZOa^h + AOab by bb. 

Ans. 5a4 — 6a2 H- 8a. 

From the 3d remark of Art. 46, it appears that the teirj. of 
the dividend containing the highest power of the leadin^j letter 
and the term containing the lowest power of the sam»», letter 
are both derived, without reduction, from the multiplica^Jun of a 
term of the divisor by a term of the quotient. Therefoie, nothing 
prevents our commencing the operation at the right Instead of 
the left, since it might be performed upon the terms containing 
the lowest power of the letter, with reference to which the ar- 
rangement has been made. 

Lastly, so independent are the partial operations required by 
the process, that after having subtracted the product of the divi- 
sor by the first term found in the quotient, we could obtain 
another term of the quotient by arranging the remainder with 
reference to some other letter and then proceeding as before. 

If the same letter is preserved, it is only because there is no 
reason for changing it ; and because the polynomials are already 
arranged with reference to it. 

OF FACTORING POLYNOMIALS. 

59. When a polynomial is the product of two or more factors, 
it in often desirable to resolve it into its component factors. 
This may oflen be done by inspection and by the aid of the 
formulas of Art. 47. 

When one factor is a monomial, the resolution may be effected 
by wi'iting the monomial for one factor, and the quotient arising 



CHAP. II.] DIVISION. 51 

from the division of the gi7en polynomial Dy this factor for the 
other, factor. 

1. Take, for example, the polynomial 

ab -\- aCj 
ir. which, it is plain, that a is a factor of both terms : hence 
ab ■}- ac = a (b -\- c). 

2. Take, for a second example, the polynomial 

ab'^c + oab^ + ab'^c^. 

It is plain that a and b^ are factors of all the terms : hence 

ab^c + 5aZ.3 + aiV = ab^ (c -}- 56 + c^). 

3. Take the polynomial 25a* — SOa^b + Iba^b^ ; it is evident 
that 5 and a^ are factors of each of the terms. We may, there- 
fore, put the polynomial under the form 

5a2 (5a2 — Qab + 362). 

4. Find the factors of Sa'^b + 9o?c + \%aHy. 

Arts. 3a2 (6 -f 3c + Qxy). 

5. Find the factors of ^a'^cx — l^acx"^ + 2ac^y — Z^a^c^x. 

Ans, 2ac {4:ax — 9x^ + c*y — I5a^c^x). 

6. Find the factors of 24ca^^cx — S0a%^c^7/ + SQa'b^cd + 6abc, 

Ans. Qabc {4:abx — ha}b^c^y + ^(i%'^d +1). 

By the aid of the formulas of Art. 48, polynomials having 
certain forms may be resolved into their binomial factors. 

1. Find the flictors of o?- -f 2a6 + 62. 

Ans. (a 4- 6) X (a + 6) 

2. 49:c* + 56a;3y + 16a:2y2 ^ (7^2 + 4^;^) (7a;2 4. 4^:^). 

3. Find tlie factors of a^ — 2ab + 62. 

Ans. (a — 6) X (a — 6). 

4 64a262c2 - 48a6c2c^ -|- Oc^d* = (8a6c - 3ci2) (Sabc - 8crf2). 
5. Find the factors of a2 _ 52. ^,^5. (a + 6) X (a — 6). 

ICm^c'i-dd^ = (4ac 4- 3(/2) (4ac _ 3^2). 



52 



ELEMENTS OF ALGEBRA 



iCHAF. U. 



GENERAL EXAMPLES. 



1. Find the factors of the pol}Tiomial Q)a% -f Sa-6^ — \(jaV 

2. Find the factors of the poljnomial ISaSc^ — 'Shc^ -f- Oa^iV 

3. Find the factors of the polvnomial 25a^5c^ — ?»Oa%c^d 
— 5ac* — 60gc6. 

4. Find the factors of the polynomial 42a-52 _ '^ahcd -\- lahd 

Ans. 7ab [6ab — cd -{- d). 

5. Find the factors of the polynomial n^ -f 2n^ + n. 
First, ?i3 + 2;i2 + 71 = 71 (?i2 -|- 2^1 + 1) 

= n{n +1) X (« + 1) 
= ?i{n + 1)2. 

6. Find the factors of the polynomial 5a-bc + lOcib^c + loabc^. 

A?is. 5abc {a + 2b -{- Sc). 

7. Find the factors of the polynomial a'^x — x^. 

Ans. X {a -\- x) (a — X). 

60t Among the different principles of algebraic division, there 
y« one remarkable for its applications. It is enunciated thus : 

The difference of the same powers of any two qv^iniitits is trxactly 
divisible by the difference of the quantities. 

Let the quantities be represented by a and 5 ; acd let m de 
note any positive whole number. Then, 

a"» — b^ 
will express the difference bet^veen the same powers of a azid 6, 
ond it is to be proved that a'" — b'^ is exactly divisible bj a — 6. 

If we begin the iivision of 

Qin — Im \,j ^ __ l^ 

we have 

\a-h 



Qm — Jm 

a^ — a"*~^5 



Ist rem. - - - 
or, by factoring 



^j»r— 1^ — 5"* 



CHAP. II.] DIVISION. 53 

Dividing a"* by a the quotient is a'*"^, by the rule foi tho 
exponents. The product of a — b by a^~^ being subtracted from 
the dividend, the first remainder is a^~'^b — 6"', which can be 
put under the form, 

b (a'"-i — b"^-^). 

Now, if the factor 

(a'"-i — i'"-!) 
of the remainder, be divisible by a — b, b times (a"''"^ — b'^^^)^ 
must be divisible by a — 5, and consequently a" — b^ must 
also be divisible by a — b. Hence, 

If the difference of the same powers of two quantities is exactly 
divisible by the difference of the quantities^ then^ the difference of 
the powers of a degree greater by 1 is also divisible by it. 

But by the rules for division, we know that a^ — h^ is divis 
ible by a — b\ hence, from what has just been proved, a? — h^ 
must be divisible by a — b^ and from this result we conclude 
that a* — b^ is divisible by a — 6 and so on indefinitely : hence 
tlie proposition is proved. 

61. To determine the form of the quotient. If we continue 
tlie operation for division, we shall find a^~'^b for the second 
term of the quotient, and a^~%'^ — b^ for the second remainder ; 
also, a^~W for the third term of the quotient, and a^~%^ — i" 
for the third remainder; and so on to the m*^ term cf the quo- 
tient, which will be 

and the m*^ remainder will be 

Qm~mJ^m — Jot Qp Jm — l^ -— Q^ 

Since the operation ceases when the remainder becomes 0, we 
sha.l have m terms in the quotient, and the result may be writr 
ten thus : 

gm — yn 



-b 



= a*^! 4- a'^25 _^ a'^-3^2 ^ ^ a6"»- -\- i*"^ 



CHAPTER m. 

OP ALGEBRAIC FRACTIONS. 

62t An ALGEBRAIC FRACTION is ai expressiou of one or more 
equal parts of 1. 

One of these equal parts is called the fractional unit. Thus, 

— is an algehraic fraction, and expresses that 1 has been divided 

into h equal parts aad that a such parts are taken. 

The quantity a, written above the line, is called the numer- 
ator ; the quantity 5, written below the line, the denominator ; 
and both are called term^ of the fraction. 

One of the equal parts, as — , is called the fractional unit ; 

and generally, the reciprocal of the denominator is the frac- 
tional unit. 

The numerator always expresses the number of times that the 
fractional unit is taken ; for example, in the given fraction, the 

fractional unit -r- is taken a times. 
6 

63. An entire quantity is one which does not contain any 
fractional terms ; thus, 

a^h 4- ex is an entire quantity, 

A mixed quantity is one which contains both entire and frao> 
tional terms ; thus, 

a% -\ — - is a mixed quantity. 

Every entire quantity can be reduced to a fractional form 
having a given fractional unit, by multiplying it by the denomi- 
nator of the fracticnal unit and then writing the product over the 
denominator ; thus, the quantity c may be reduced to a fractional 



CHAP. III.] ALGEBRAIC FRACTIONS. 55 

form with the fractional unit — , by multiplying c by 6 and 

he 
dividing the product by J, which gives — . 

64. If the numerator is exactly divisible by the denominator, 
a fractional expression may be reduced to an entire one. by sim- 
ply performing the division indicated; if the numerator is not 
exactly divisible, the application of the rule for division will 
sometimes reduce the fractional to a mixed quantity. 

65. If the numerator a of the fraction — be multiplied by 
any quantity, q, the resulting fraction — will express q times 
fts many fractional units as are expressed by — ; hence: 

Multiplying the numerator of a fraction hy any quantity »a 
equivalent to multiplying the fraction by the same quantity. 

66. If the denominator be multiplied by any quantity, q^ the 
value of the fractional unit, will be diminished q times, and the 

resulting fraction — will express a quantity q times less than 

tlie given fraction ; hence : 

Multiplying the denominator of a fraction hy any quanUty^ is 
equivalent to- dividing the fraction hy the same quantity, 

67. Since we may multiply and divide an expression by the 
same quantity without altering its value, it follows from Arts 
05 and GQ, that : 

Both numerator and denominator of a fraction may he multiplied 
by the same quantity^ without changing the value of the fraction. 

In like manner it is evident that : 

Both numeratcr and denominator of c fraction may he divided 
hy the same quantity without changing ihe value of the fraction. 

68. We shall now apply these principles in deducing rules 
for tho transformation or reduction of fractions. 



56 ELEMENTS OF ALGEBEA. ICHAP. HL 

I. A fractional is said to be in its simplest form when the numer. 
ator and denominator do not contain a common factor. Now, 
since both terms of a fraction may be divided by the same 
quantity without altermg its value, we have for the reduction 
of a fraction to its simplest form the followmg 

RULE. 

Resolve both numerator and denomhiator intc their simple fao- 
tors (Art. 59) ; then, suppress all the factors common to both 
terms, and the fraction ivill be in its simplest form. 

Remark. — When the terms of the fraction cannot be resolved 
into their simple factors by the aid of the rules already given, 
resort must be had to the method of thft greatest common divi 
sor, yet to be explained. 

EXAMPLES. 

IT.! ^ o . 3a5 4- 6ac . . , ^ 

1. Keduce the fraction ^ — ; — -- tc f^ simplesl form. 

We see, by inspection, that 3 and a f*^' fpcto^' of thr nu- 
merator, hence, 

3a5 + Qac = ^a{b + 2c) 

We also see, that 3 and a are factors \^ ihe <\2'iomii:af •««, 

hence, 

Zad+ 12a = 3a(c^ + 4). 

Zab + 6ac _ 3a (5 + 2c) _ b -\- 9r 
"^^^ Zad+ 12a "" 3a {d + 4) ~ J+Il' 



,. ^ , 6a25 + 3ac . . ., , 

2. Keduce r— ; — -—z — -, to its simplest form. 
9a6 + 3ai -^ 



2ch f c 



„ ^ , 256c +5^/ . . -, ^ 

3. Keduce . ^^,^ . ^^, to its simplest form. 

3562 ^ 15^ -i 

A ^'^f 

7/ -fS 

^ _ _ 54a6c . . , ^ 

4, Keduce tt-t, — —r: — r to Its simplest form. 

45a2c + 9aca ^ 

Ans. ,; — -— 
oa -h cf 



CHAP. III.] 



ALGEBRAIC FRACTIONS. 



67 



b. Reduce 



0. Reduce 



7. Reduce 



84tt62 



12acd — 4cd^ 
12cd/ + 4c^d 



18aV - Sacf 
27ac2 —ijac-i 



to its simplest furm. 



^i/ 



76 



o its simplest form. 



Ans. 



Za — d 



.0 its simplest form. 



Ans. 



9c — 2c2* 



11. From what was shown in Art. 63, it follows that we may 
reduce the entire part of a mixed quantity to a fractional form 
wich the same fractional unit as the fractional part, by multiply- 
ing and dividing' it by the denominator of the fractional part. 
The two parts having then the same fractional unit, m.ay be 
reduced by adding their numerators and writing the sum obtained 
over the common denominator. 

Hence, to reduce a mixed quantity to a fractional form, we 
have the 

RULE. 

Multiply the entire part hy the denorninator of the fraction: 
then add the product to the numerator and write the sum over iht 
denominator of the fractional part. 

EXAMPLES. 

1. Reduce x — - '- to the form of a fraction. 



Here, 



a2 _ ^2 ^2 _ (^2 _ ^2) 2a;2 — a^ 



U. Reduce x to the form of a fraction. 

la 



Ans. 



2a 



58 



3. Reduce 



i. Reduce 1 



ELEMENTS OF ALGEBRA. 
2a;- 7 



[CHAP lU 



Sx 



X — a 



to the form of a fraction. 



Ans. 



Zx 



5. Reduce 1 -f 2:c 



6. Reduce 3.r — 1 



bx 



to the form of a fraction. 

An. ""-- + ^ 
a 

to the form of a fraction. 



^/?5. 



5a; 



3a -2 



to the form of a fraction, 

9aa; — 4a — 7a; + 2 



^?2S. 



:6a 



Remark. — We shall hereafter treat mixed quantities as though 
they were fractional, supposing them to have been reduced to a 
fractional form by the preceding rule. 

III. — From Art. 64, we deduce the following rule for reducing 
a fractional to an entire or mixed quantity. 

RULE. 

Divide the numerator by the denominator^ and continue the oper 
ation so long as the first term of the remainder is divisible by the 
first term of the divisor : then the entire part of the quotient found, 
added to the quotient of the remaifider by the divisor, will be the 
mixed qua'itity required. 

If the remainder is 0, the division is exact, and the quotient 
is an entire quantity, equivalent to the given fractional expres- 
sion. 

EXAMPLES. 

_ , ax -{- a^ . , 

1. Reduce to a mixed quantity. 



Ans. z=a -{- 



CHAP. III.J ALGEBRAIC FRACTIONS. 59 

ax — x^ 
2. Reduce to an entue or mixed quantity. 



Reduce ' — r to a mixed quantity. 



4. Reduce to an entire quantity. 



/J.3 y3 

5. Reduce to an entire quantity. 



-471?. a — IT. 



2a». 
6 



Ans. a -f X, 



x — y 



Ans. a;2 -f rry + y^. 



^ ^ ^ 10:c2 _ 5.p _|_ 3 

6. Reduce to a mixed quantity. 

ox 

Ans. 2x — 1 -\- - -. 
bx 

IV. To reduce fractions having different denominators to equiv 
alent fractions having a common denominator. 

Let — , — and —, be any three fractions whatever. 

0(1 J 

It is evident that both terms of the first fraction may be mul 

tiplied by df giving ■—,-, and that this operation does not 

change the value of the fraction (Art. 67). 

In like manner both terms of the second fraction may be 

' . . . . hcf 

multiplied by 5/, giving — -- ; also, both terms of the fraction 

-— may be multiplied by 6a, givmg -r-j . 
J J 

T^ . - , - . odf hcf , hde 

11 now we examine the three fractions — -, -^ and 7-7^ 

odf odf hdf 

we see that they have a common denominator, hdf and that 

each numerator has been obtained by multiplying the numerator 

of the correspc nding fraction by the product of all the denom- 

Inators except its own. Since we may reason in a similar 

manner upon any fractions whatever, we have the following 



60 ELEMENTS OF ALGEBRA. [CHAP. HI. 

RULE. 

Multiply each numerator into the p:oa.ict of all the denominth- 
tors except its own^ for new numerators^ and all the denominators 
together for a common denominator. 

• 

EXAMPLES. 

1 . Eeduce — and — to equivalent fractions having a com 

mon denominator. 

a X c zzz ac 



y the new numerators. 



b Xb=b^ 
and - b X c = be the common denominator. 

2. Reduce — and to equivalent fractions having ;^ ooiu 

c 

mon denominator. Ans. rr- and . 

be be 

Sx 25 

3. Reduce — -, -— - and d, to eouivalent fractions having a 

2a 3c ' - ^ 

^ . . 9cx 4ab ^ Gacd 

common denommator. Ans. — — , -- — and . 

bac bac bac 

3 2t 2x 

4. Reduce — , ~ and a -\ , to equivalent fractions hav- 

4 o a 

9a Sax , 12^2 + 24a; 

ma a common denommator. Ai^s. -— , -— — and — : . 

12a 12a 12a 

1 a^ a^ -|- x^ 

5. Reduce -—, — - and , to equivalent fractions hai>t 

2 6 a -f- x 

iiig a common denominator. 

8a + Sx 2a3 + 2a^x , ba^ -^ bx^ 
" * 6a + bx' ba 4- bx ba + bx ' 

6. Reduce -, , and — , to equivalent fractions bay 

a — b ax c 

ing a common denominator. 

a^cx ac^ — abc — bc"^ -f cb"^ -, «^5^ — ^^^-J? 

a\x — abcx^ a^cx — abcx ahx - abcx 



CHAP. III. J ALGEBRAIC FRACTIC^S. 61 

V. To add fractions together. 

Quantities cannot be added together unless they have the 
same unit. Hence, the fractions must first be reduced to equiv- 
alent ones having the same fractional unit ; then the sum of 
tlie numerators will designate the number of times this unit 
is to be taken. We have, therefore, for the addition of frao. 
tions the following 

RULE. 

Reduce the fractions^ if necessary^ to a common denominator : 
then add the numerators together and place their sum over the 
common denominator. 

EXAMPLES. 

1. Find the sum of -— , — - and — r. 

b d f 

Here, - a X d xf = adf \ 

c X h X / = chf > the new numerators. 

e X b X d = ebd) 
And - b X d xf = bdf the common denominator. 
^j. adf , cbf , ebd adf + cbf + ebd , 
^^^"^*'' 4^Mf+Wf=-^-^ *'^^"'"- 

2. To a-^4 add 6 + ^-^. Ans. « + 6 + ?^^1^^. 

c be 

3. Add — , — - and — together. Ans. x -\- -^, 

A \;j;i ^ —^ 1 4a: ^ . ^ 19.r — 14 

4. Add — -— and — together. Ans. . 

K HAA 1^-2^ o , 2ar-3 . ^ , lO.r-17 

5. Add X H -— to Sx H — . Ans. Ax -\ — . 

6. It is required to add 4a;, -— and ^-J— together. 

2a 2x ° 

. ^ . 6x^ -{- ax -\- a^ 

Ans. 4x H . 

2ax 

2x Ix 2x -\- \ 

7. It IS required to add — - -— and — together. 

49a; + 12 
Ans. 2x t- eO ' 



62 ELEMENTS OF ALGEBRA. [CHAP. ITL 

1x X 

8. Ifc is reCj^uired to add 4a;, — and 2 4-^ together. 

. . 44a; + 90 

Ans. 4:X -\ . 

45 

2a; 8x 

9. It is required to add Sx -\- — and x — — together. 

Ans. ox + ——. 
45 

10. What IS the sum of ;, — — r and 



Ans. 



a — V a -\- b a -\- x' 

a^ — ox"^ -\- a^b — bx"^ + aP'C + acx — ahc — bcx -\- a^d — b'^d 



d^ — h'^a -\- a?x — b'^x 
_ a^ + a^{b + c -\-d) — a [x^ — ex -}- be) — b {x^ + ex + hd) 
a^ -j- d^x — ab'^ — b'^x 

VI. To subtract one fraction from another. 

Reduce the fractional quantities to equivalent ones, having the 
same fractional unit ; the difference of their numerators will 
exgress how many times this unit is taken in one fraction more 
than in the other. Hence the following 

RULE. 

I. Reduci the fractions to a common deiiominator, 

II, Subtract the numerator of the subtrahend from the numer- 
ator of the minuendy and place the difference ovei' the common 
denominator. 

EXAMPLES. 

, ■ X — a 2a — 4a; 

1. rrom - - - — — — subtract — . 

26 3c 

^^ (x— a) X 3c = Sea; — 3ac ) 

i^^^^i /,X . { o7 AT. o7 r the numerators. 
(2a — 4a:) X 26 = 4a6 — 86a; ) 

And, 26 X 3c = 66c the common denominatci 

3ca; — 3ac 4a6 — 86a; 3ca; — 3ac — 4a6 + 86ir 



Hence, 



66c 66c 66c 



^ -r. 12a; 1.^ 3a; , 39jr 

2. From - - —— subtract —-. An*, — — . 

7 9 «»5 



CHAP. III.] 


ALGEBRAIC FKACTIONS. 




a From - 


- 5y 


subtract -^. A as. 


Sly 

8 • 


4. From - 


Zx 

7 


2x 
subtract — -. A is. 

y 


03' 


• 
5. From - 


X -\- a 
b 


c , dx ■\- ad 

subtract — . Ans. 7-- 

d bd 


~6c 


G. From - 


Zx-\- a 
56 


2:r + 7 
subtract — - — . 









24^ + 8a — 106a; - 


-356 




^'''- 406 




7. From - 


- 3^ + i 






c 








cx~\-hx 
Ans. 2x H -— 


-ab 



63 



6c 



VIJ. To multiply one fractional quantity by another. 

a, c 

Let - represent any fraction, and - any other fraction; and 

Jet it be required to find their product. 

If, in the first place, we multiply - by c, the product will 

be — , obtained by multiplying the numerator by c, (Art. G5) ; 

-but this product is d times too great, si^ce we multiplied 

- by a quantity d times too great. Hence, to obtain the true 

product we must divide by d, which is effected (Art. 66) by 
multiplying the denominator by d. We have then. 

a c ac . 

b''d = bd^ ^^"^^ 



RULE. 

I. Cancel all factors common to the numerator and denomi- 
nator. 

II. M.dtiply the numerators together for the numerator of the 
product, and the denominators together for the denominator of the 
l/i-oduct. 



(54 ELEMENTS OF ALGEBRA. [CHAP. IIL 

EXAMPLES. 

I, Multiply a -I by — . 

-r^. ^ , bx a^ -\- hx 

t irst, - - - - a-\ = ; 

a a ' 

-TT «^ + bx c a?c 4- bcx 
Hence, - . X — — • 



d 



9a» 



2. Required the product of -^ and — . Ans 

J5. Required the product of --- and — -. Ans, — . 

5 2tt 5a 

4. Find the continued product of — , and — — -. 

a c 2b 

I Ans. 9ax 

bx CL 

5. It is required to find the product of 6 H and — . 

. ab ■\- hx 

Ans. . 

x 

^2 52 x"^ -\- b"^ 

6. Required the product of — 7 and — . 

x^ — b^ 

/J. _{_ 1 X 1 

7. Required the product of a; ^ — and — x~a* 

ax^ — ax -j- x^^""^ 1 

Ans. r- — . 

a^ + ao 

fix Q! X 

8. Required the product of a ^ and — 77^* 

a "~ X X "T" X 

Ans. ' ■ — ( 



CHAP. III.] ALGEBRAIC FRACTIONS. 65 



VIII. To divide one fraction by another, 

Let — represent the first, and 
he division may be indicated thus 



Let — represent the first, and — the second fraction; then 



(i> 



If now we multiply both numerator and denominator of this 
complex fraction by — , which will not change the value of the 

fraction (Art. 67), the new numerator will be r— , and the new 

be 

denominator —, which is equal to 1. 



a c ^\b ) _\ be; ad 

' ' " 6 ' d ~ I c\ ~ 1 ~~ be' 



(t) © 



This last result we see might have been obtained by inverting 
llie terms of the divisor and multiplying the dividend by the 
resulting fraction. Hence, for the division of fractions, we have 
the following 

RULE. 

Invert the terms of the divisor and multiply the dividend hy the 
resulting fraction. 

EXAMPLES. 



1. Divide 



Hence a - - -- I- - ^^' ~ ^ x -^1 - ^^^^ "" ^^ 
Hence, a ^c " ^ " 2c ^ / " 2c/ * 

*Z. Let — be divided by — . Ans. — -. 

5 



[] ^Y^T- 



^^r^ 



^^ 


ELEMENTS OF ALGEBRA. 


[CHAP. m. 


3. Let 


4a;2 

— - be divided by ^x. 




A ^ 


4. Let 


— - — bo divided by — -. 




Ans. ^+\ 
Ax 


5. Let 


X X 

be divided by -— . 




. 2 

Ans. -. 

X — I 


6. Let 


T ^ ^'''^^^ ^^' S- 




bbx 
Ans, ^. 


7. Let 


—-— r- be divided by -— -. 




x-b 

Ans. ^ - . 


8. Let 


x^ — h^ 

— — — — — be divided 'by 

x^ — 2bx -{- b^ ^ 


x^ 

X 


-\-bx 

Ans. x-\ . 

X 


9. Divide — — — by -. Ans. 

\ — X -^ I — x^ 


ax{l -{-x) —x — \ 
a 


10. Divide "+J by ]+\. 
a — 1 '' 1 — a2 




Ans. - (1 4 a). 


69. If 


we have a fraction of the form 
a 




we may 


observe that 






— G 

1 


- = — c, also - = —c and 
> — b 


— 


y = c ; that is, 



77ie sign of the quotient will be changed by changing the sign 
either of the numerator or denominator, but will not be affected by 
changing the signs of both the terms. 

70. We will add two propositions on the subject of fractions. 

I. If the same number be added to each of the terms of a proper 
fraction, the fraction resulting from these additions will be greater 
than the first ; but if it be added to the terms of an imj^ropet 
fraction, the resulting fraction will be less than the first. 

Let the fraction be expressed by — . 

Let m represent the number to be added to each terra : thef 

. /. . -n T a -\- m 

the new fraction will be, . 

+ m 



CHA.P III.] ALGEBRAIC FRACTIONS. 67 

In order to compare the two fractions, they must he reduced 
to the same denominator, which gives for ^ 

a a6 + am 



the first fraction, 

and for the new fractio^j, 



6 b'^-\-bm 
a-\- m ab -\-bm 



b -\- m 6'^ + bm 

Now, the denominators being the same, that fraction will be 

the greater which has the greater numerator. But the two 

numerators have a common part ab, and the part bm of the 

second is greater than the part am of the first, when 6 > a : 

hence 

ab -\- bm "^ ab -\- am ; 
that is, when the fraction is proper, the second fraction is greater 
than the first. 

If the given fraction is improper, that is, if a > ^, it is plain 
that the numerator of the second fraction will be less than that 
of the first, since bm would then be less than am. 

II. If the same number be subtracted from each term of a proper 
fraction, the value of the fraction will be diminished ; but if it be 
subtracted from the terms of an improper fraction, the value of the 
fraction will be increased. 

Let the fraction be expressed by — , and denote the number 

to be subtracted by m. 

Then, -; will denote the new fraction. 

b — m 

By reducing to the same denominator, we have, 

a ab — am 



md 



a — ra ab — bm 



b — m 6^ — bm 
Now, if we suppose a < b, then am < bm; and 'S am < bm^ 

then will 

ab — am "^ ab — bm : 

that is, the new fraction will be less than the first. 

If a > 6, that is, if the fraction is improper, then 

am > bm, and ab — am <^ ab — bm, 

that is, the new fraction will te greater than the first. 



68 ELEMENTS OF ALGEBRA. LCHAP. IH. 



GENEflAL EXAMPLES. 

1. Add T-^ to T-T— ^. Ans. -^ -f. 

11 .2 

2. Add -— — to . Ans. 



]fa; l — x \-^ 

3. From — —- take — — r. Ans. 



a — b a-\-b' o? — H 

4. From take -— — r. ^ns. 



1 _ a;2 1 -I- ic** 1 — ar* 

. x^-nx + 2^ 

-]jUi>,.y) ^^^- J2 • 

« ,r -, . -, ic* — 6* , x"^ ■\-bx .' , , T, 

6. Multiply Jj:f.-2j^^j5 by --^. ^n,. ^3 + 53^. 

«,-r^. .-, a + ^ « — ^t a -\- X a — x 

7. Divide — 1 ; — by ■ — ; — . 

a — X a -{- X a — x a -^ x 

a? + x^ 

2ax ' 



Ans. 



8. Divide 1 H — ~ oy 1 — -. Ans. n. 

n-\-\ n -{-\ 



EXAMPLES INDICATING USEFUL EORMS OF REDUCTION. 

adfx^ + hcfx 4- ^de 
~ bdfa^ 

^ ^ X ^ ^ 9 _ aclfhx^ hcfhx^ hedhx'' bdfg^ 

Jx^ dx^ ~fx^ ~ Kx^~ bdfhx^^ "^ bdfhx^^ ~ bdfhx^o ~ bdfhx^^ 
adflix^ -\- bcfhx^ — beihx — hdfg 
"^^ b^x'^ ■ 



JH 


kP. III. EXAMPLES IN FRACTIONS. 60 


1. 


l + x^ 1 _ a:2 
" 1 - a;2 ' 1 + 0:2 


(1 4- x-^Y + (1 - ^')' 
" (1 - X') (1 4- a-2) 
2(1 4- X') 


2. 


I + a; ' 1—x 


1-x l+x 


-(14-^)(1-^) (l+x){l^x) 
1 -0:4- 1 +^ 






-(14-^)(1-^) 
2 




- l~a:2- 


8. 


a -\- b a — b 
a — b a -\- b 


(^4-^,)2_(^_5)2 

- {a-\-b){a-b) 

4:ab 
~" a2 — b"^' 


4. 


1 -f a;2 1 - a;2 
I — a;2 1 + a;2 


(1 4- a;2)2 (1 _ x^Y 

- {l-x^){l-\-x^) (l-o;2)(14-o:2) 

(1 4- x^Y - (1 - ^^Y 

- (l-o;2)(14-a;2) 
4o;2 

~1 -a:** 


5. 


1 _f. a:2 1 _ a:2 

1 - a;2 • 1 4- a;2 


_ 1 4- ^2 1 + ^2 (1 + x^Y 
- 1 _ ^2 ^S - a;2 ~ (1 — x^Y' 


6. 


ic* — 6* . a;2 + ?>o; a:* — 5* ^ x — b 
x^ — 2bx-^b^' x—b ~ x'^ — 2bx -i- b"^^ x^-bx 






(x^-b^){x-b) 




~ (2-2 _ 2bx 4- ^2) (o;2 4- bx) 
{X^ - 62) (o:2 4- ^2) (^ _ 5) 

- (o: - ^*)2 o; {x + b) 

(x + b){x-b){x^-\-b^){x-b) 
X (x -b)(x- b) (x + b) 

««4-62 



70 ELEMENTS OP ALGEBRA. LCHAP. IIL 

Of the Symbols 0, oo and — . 

71. The symbol is called zero, which signifies in ordinary 

language, nothing. In Algebra, it signifies no quantity : it is 

also used to expres a quantity less than any assignable quantity. 

Tlie symbol go is called the symbol for infinity ; that is, it is 

used to represent a quantity greater than any assignable quaiUity. 

If we take the fraction — , and suppose, whilst the value of 

a remains the same, that the value of b becomes greater and 
greater, it is evident that the value of the fraction will become 
less and less. When the value of b becomes very great, the 
value of the fraction becomes very small ; and filially, when b 
becomes greater than any assignable quantity, or infinite, the 
value of the fraction becomes less than any assignable quantity, 
or zero. 

Hence, we say, that a finite quantity divided by infinity is 
equal to zero. 

We may therefore regard — , and 0, as equivalent symbols. 

If in the same fraction — , we suppose, whilst the value of a 

remains the same, that the value of b becomes less and less, it 
is plain that the value of the fraction becomes greater and 
greater; and finally, when b becomes less than any assignable 
quantity, or zero^ the 'alue of the fraction becomes greater than 
any assignable quantitj, or infinite. 

Hence, we say, that a finite quantity divided by zero is equal 
to infinity. 

We may then regard — and :xi as equivalent symbols : Zerc 

and infinity are reciprocals of each other. 

The expression — is a symbol of indetei-mination ; that is. it 

is employed to designate a quantity which admits of an infinite 
number of values. The origin of the symbol will be explained 
in the next chapter. 



T 



K 



CHAP, iii.j algebeaic fractions. 71 

It should be observed, however, that the expression ~ is not 

always a symbol of indetermination^ but frequently arises from 
the existence of a common factor^ in both terms of a fraction, 
which factor becomes zero, in consequence of a particular hypo- 
thesis. \ 

1. Let us consider the value of x m the expression 

O? — 62 

M^ in this formula, a is made equal to 6, there results 



But, - - . a^ — h'^^(a — h){a^-^ah-\- IP) 
and - . a2 — 62 = (a - 6) (a + 6), 
hence, we have, 

_ (« - h) (a2 + a6 4- ^2) 
* - ■ (a - 6) (« 4- h) • 

Now, if we suppress the common factor a-r-h^ and then sup 
pose a z=zh, we shall have 

3a 

2. Let us suppose that, in another example, we have 

_ a^ — 62 
* - (a - bY 

If we suppose az=.h^ we have 



If, however, we suppress the factor common to the numerator 
and denominator, in the value of x^ we have, 
_ (a + ^>) ja-b) a -\-b 
^ ~ (a - 6) (a, _ 6) ~ a - 6" 

'If now we make a = b, the value of x becomes 

26 



ELEMENTS OF ALGEBRA. 



[CHAP. IIT. 



3. Let us suppose in another example, , 



X = 



^rzrp-' 7 o.^OC^'^ V A / i^A; 



in which the value of x becomes —- when we make a =_ i, 
f 

It* we strike out the common factor a — h, we shall find 



r: o^ 



Kion 



, it is necessary to ascertain w^hether it does not arise 



a^ -\- ab -i- b^ 
If now we make a z=b, the value of x becomes 

Therefore, before pronouncing upon the nature of the expres 

from the existence of a common factor in both numerator and 
denominator, which becomes under a particular hypothesis. 
If it does not arise from the existence of such a factor, we 
conclude that the expression is indeterminate. If it does arise 
from the existence of such a factor, strike it out, and then make 
tiie particular supposition. 

If A and B represent finite quantities, the resulting value of 
llie expression will assume one of the three forms ; that is : 
A A 

1^ T ""' a'^ 
it will be either finite^ infinite, or zero. 

This remark is of much use in the discussion of problems. 



CHAPTER IV. 

EQUATICNS OF THE FIRST DEGREE INTOLVING BUT ONE UNKNOWN QUANTlTTi 

72i An Equation is the algebraic expression of equality bo- 
tween two quantities. 

Thus, X = a -}- b, 

is an equation, and expresses that the quantity denoted by x i» 
equal to the sum of the quantities represented by a and b. 

Every equation is composed of two parts, connected b} the 
sign of equality. The part on the left of this sign is called the 
first member^ that on the right the second member. The second 
member of an equation is often 0. 

73« An equation may contain one unknown quantity only, or 
it may contain more than one. Equations are also classified 
according to their degrees. The degrees are indicated by ihs 
exponents of the unknown quantities which enter them. 

In equations involving but one unknown quantity^ the degree is 
denoted by the exponent of the highest j^owcr of that quantity in 
any term. 

In equations involving more than one unknown quantity^ thn 
degree is denoted by the greatest sum of the exponents of the unknown 
quantities in any term. 

For example: 
dx -'r b z::^ ex -{- d 
az -[-2>by -\-cz-\-M—0 

ax^ -i-2bx -{-€ = 

ax^ -{-bxy i-cy^-\- d = 

<^x^ -\- '^dgx'^ rr: abx — c^ 
4a.ry2 — 2cy^ + «^^y = 3 
and so on. 



[- are equations of the first degree. 
[• are equations of the second degree. 
- are equations of the third degree, 



74 ELEMENTS OF ALGEBRA. ICHAP. IV 

74. Equations are likewise distinguished as numerical equations 
and literal equations. The first are those which contain numbers 
only, with the exception of the unknown quantity, which i«* 
always denoted by a letter. Thus, 

are numerical equations. 

A literal equation is one in which a part, or all of the known 
■quantities, are represented by letters. Thus, 

bx- ~h ax — Zx — 5, and ex + dx"^ — c -\- f^ 
are literal equations. 

75i An identical equation is an equation in which one member 
is repeated in the other, or in which one member is the result of 
certain operations indicated in the other. In either case, the 
equation is true for every possible value of the unknown quan- 
tities which enter it. Thus, 



,cx -\-h := ax-{- b^ {x -{- ay =zx^-^ 2ax -f a^, — — — =:x — y, 
are identical equations. 

76 1 From the nature of an equation, we perceive that it must 
possess the three following properties : 

1st. Tlie two members must be composed of quantities of the 
same kind. 

2d. The two members must be equal to each other. 

3d. The essential sign of the two members must be the same, 

76.* An axiom is a self-evident proposition. We may here 
enumerate the following, which are employed in the transformct- 
tion and solution of equations : 

1. If equal quantities be added to both members of an equation, 
the equality of the members will not be destroyed. 

2. If equal quantities be subtracted from both members of an 
equation, the equality ^^11 not be destroyed. 

3. If both members of an equation be multiplied by equal 
quantities, the products will be equal. 

4. If both members of an equation be divided by equal quan 
titles, the quotients will be equal. 

5. Like powers of the two members of an equation are equal 
C. Like roots of the two members of an equation are equal 



CHAP. IV.] EQUATIONS OF THE FIRST DEGREE. 75 

Solution of Equations of the First Degree, 

77. The solution of an equation is the operation of finding a 
value for the unknown quantity such, that when substituted for 
tne unknown quantity in the equation, it will satisfy it ; that is, 
make the two members equal. This value is called a root of 
the equation. 

In solving an equation, w^e make use of certain transformations, 

A transformation of an equation is an operation by which we 

change it^ form without destroying the equality of its members. 

First Transformation, 

78. The object of the first transformation is, to reduce an 
equation, some of whose terms are fractional, to one in v)liich all 
of the terms shall be entire. 

Take the equation, 

First, reduce all the fractions to the same denominator, by the 
known rule ; the equation then becomes 
48.r 54a; 12.r _ 

72 ~ "72 "^ "72 ~ • 

i? now, both members of this equation be multiplied by 72, 
the equality of the members will be preserved (axiom 3), and 
the common denominator will disappear ; and we shall have 

48^ — 54^ + 12^ == 792 ; or by dividing 
both members by ^, %x ~ 9^ + 2a? = 132. 

The last equation could have been found in another manner 
by employing the l6ast common multiple of the denominators. 

The common multiple of two or more numbers is any num- 
ber which each will divide without a remainder ; and the least 
common multiple, is the least number which can be so divided. 

The least common multiple of small numbers can be found 
by inspection. Thus, 24 is the least common multiple oi 4, 6 
and 8; and 12 is the least common multiple of 3, 4 and 6. 



76 ELEMENTS OF ALGEBRA. [CHAP. IV. 

Take the last equation, 

We see that 12 is the least common multiple of the de- 
nominators, aud if we multiply each term of the equation by 
12, reducing at the same time to entire tezms, we obtain 

8:r — 9a; + 2a; = 132, 
tlie same equation as before found. 

Hence, to transform an equation involving fractional terms to 
one involving only entire terms, we have the following 

RULE. 

Form the least common multiple of all the denominators^ and 
then multiply both members of the equation by it, reducing fractional 
to entire terms. 

Tills operation is called clearing of fractions. 



EXAMPLES. 
X X 

1. Reduce -r^ + -7 3 = 20, to an equation involving only 

entire terms. 

We see, at once, that the least common multiple is 20, by 
which each term of the equation is to be multiplied. 
X 20 

Now, -:;- X 20 = X X -^ — 4:X, 

o 5 ' 

and —- x20 = X X —r = 5x: 

4 4 

that is, we reduce the fractional to entire terms, by multiplying 

the numerator by the quotient of the common multiple divided hy 

the denominator^ and omitting the denominators. 

Hence, the transformed equation is 

42; -F 5a; — 60 m 400. 

x X 

2. Reduce — -\- ~ 4 = 3 to an equation involving only 

entire terms. Ans. 7a; + 5a; — HO = 105. 



CHAi', IV.J EQUATIONS OF THE FIRST DEGREE. 77 

ci c 

3. Kbtiuce — —■\-f=g to an equation involving only 

entire terms. Ans. ad—hc-\- hdf ■=. hdg, 

4. Eeduoe 'ii:o equation 

QX 2c'^x ^ 4hc^x 5a^ 2c^ 
CO W 6" a 

to one involving only entire terms. 

Ans. a'bx — 2a^c^x -f 4^462 _ 4^^^'^^ _ 5^6 _^ 2a^h'^ — Ba^b\ 

Secc.id Transformation, 

79. The object of the second transformation is to change 
any term from one member of an equation to the other. 

Let us take the equation 

ax -\- b v^ d — ex. 
If we add ex to both membci!?, the equality will not be de- 
stroyed (axiom 1), and we shall IkvVC 

ax -i- ex -{- b ^ d — ex -\- cx'j 
or by reducing, ax -{- ex -{- b =^ d. 

Again, if we subtract b from both members, the equality- 
will not be destroyed (axiom 2), and we shall have, after 
reduction, 

ax -\- ex :=d — b. 
Since we may perform similar operations on any other equation, 
we have, for the change or transposition of terms, the following 

RULE. 

Any term of an equation may be transposed from one member 
to the other by changing its sign. 

80. We will now appi/ the preceding principles to the solit 
tion of equations of the first degree. 

For this purpose let us assume the equation 
a + b __ 7 _ 7 _ a -\- d 
c ~ ' a ' 

Clearing of fractions, we have, 

a{a-\- b)x — acd = abcx — c[a-{- d). 



78 ELEMENTS OF ALGEBRA. fCH^P. IV. 

If, DOW, -^e perform the operations indicated in both merahers, 
we shall obtain the equation 

Kj?x + ahx — acd = obex — ca — cd. 
Transposing all the terms containing x, to the first member, 
and all the known terms to the second member, we shall have, 
a^x + ahx — obex = acd — ac — cd. 
Factoring the first member, we obtain 

{a^ -{■ ab — abc) x = acd — ac — cd : 
If we divide both members of this equation by the oo« 
efficient of x, we shall have 

acd — ac — cd 

^ — 

d^ -\- ab — abc 
Any other equation of the first degree may be solved in a 
similar manner : 

Hence, in order to solve any equation of the first degree, 
we have the following 

RULE. 

I. Clear the equation of fractions, and perform in both members 
all the algebraic operations indicated. 

II. Transpose all the terms containing the unknown quantity to 
the first member, and all the known terms to the second member, 
and reduce both members to their siynplest form. 

ni. Resolve the first member into two factors, one of which shall 
he the unJcnoivn quantity ; the other one will he the algebraic sum 
of its several co-efiicients. 

IV. Divide both members by the co-efficient of the unknown quan- 
tity ; the second member of the resulting equation will be the re- 
quired value of the unknown quantity. 

1. Take the numerical example 

5a: _ 4ar _ 7 13a; 

12 1 ~ ~8 6~* 

# Clearing of fractions 

IOj* — 32a; — 312 = 21 — 52a;; 



CHAP. IV. J EQUATIONS OF THE FIRST DEGREE Y9 

tratsposing and reduciDg 

30.T = G33 : 
"Whence, by dividing both members of the equation by 30. 

x= 11.1. 
If we substitute this vahie of x^ for ar, in the given equation, 
it will verity it, that is, make the two members equal to each 
other. 

Find the value of x in each of the folio ving 



Ans, a: = 3. 

Ans, X— \\\. 

Ans. a: = 2, 

Ans. X =z Q. 



Ans.. = ^. 



««.««, J 6 — 3a 

6. 3aa: H 3 = oar — a. Ans. x = -. 

2 - 6a — 26 

.a; — 3. a; ^^ a;— 19 

1. = 20 . Ans. X = 23i. 

2 3 2 * 

^x-\-3,x^x— 5 , « 

8. -^ + Y = * f-- ^««- " = 3A- 

^ ax ^ b a bx bx — a . Sb 

9. -. — 1 = . Ans. X = 





EXAMPLES. 


1. 


3a: -^ 2 + 24 = 31. 




2. 


a; + 18 = 3a; — 5. 




3. 


6 ~ 2a; + 10 = 20 - 33" - 


2. 


4. 


, + |, + i., = „. 






X) 


6. 


2x - —x + 1 = 6a;- 2. 


!0 




/3 o 



4 3 2 3 3(Z — 26 

, ^ 3ar Ibx „ . cdf -\- ird 

10. 4 =/. Ans. X = \ , . 

c d '' Sad — 2bc 

,, 8a.r- 6 36 - c , , 56 + 96 — 7c 

11. =: 4 — ?i. Ans. X = » 

.7 2 16a 

,„ a; a; — 2 a; 13 

\2. — = _. ^dns. a; = la 

6 3 2 3 



80 ELEMENTS OF ALGEBRA, [CHAP. IV 

X XX X ahcdf N/ 

abed bed — acd -j- abd —abc ^ 

C ^ 14. a; — — H — — = ar + 1. Ans. a; = 6. 

lo 11 



15. Z._|_^_^ = ^12||. ^n. 



. . = 14.>(^ 



,^ ^ 4a;-2 35;-l 

16. 2^ — = — - — . Ans. X = Z. 



17. 3a; H — =ix -\- a. Ans. x 



6 4- ^ 



18. (- + ^)(--^) --3...1^J!_2. + -^-^- 



^713. X = 



a + 6 '6 

a* + Sa^ + 4a252 _ QaP + 26* 



26 (2a2 + ab- 62) 



Problems jiving rise to I'Jquaiions of the First Degree, involv' 
ing hut c%e Unknown Quantity. 



81 • The solution of a problem, by means of 
of two distinct parts — 

1st. The statement of the problem ; and 

2(1. The solution of the equation. 

We have already explained the methods of solving the equa- 
tion ; and it only remains to point out the best manner of making 
the statement. 

The statement of a problem is the operation of expressing, 
algebraically, the relations between the known ard unknown 
quantities which enter it. 

Tins part cannot, like the second, be subjected to any well- 
defined rule. Sometimes the enunciation of the problem furnishes 
the equation immediately ; and sometimes it is necessarj^ to dis- 
oover, from the enunciation, new conditions from which an equa- 
tion may be formed. 



CHAP. IV. EQUATIONS OF THE FIRST DEGREE. 81 

Tlie conditions enunciated are called explicit conditions^ aud 
those which are deduced from them, implicit conditions. 

In almost all cases, however, we are enabled to discover the 
equation by applying the following 

RULK 

Denote the unknown quantity by one of the final letters of tite 
alphabet, and then indicate, by means of algebraic signs^ the sarne 
operations on the known and unknown quantities, as would be 
necessary to verify the value of the unJxnoivn quantity, were such 
value known. 

PROBLEMS. 

1. Find a number such, that the sum of one half, one third! 
luid one fourth of it, augmented by 45, shall be equal to 448. 

Let the required number be denoted by x. 



X 
X 
X 



Then, one half of it will be denoted by - - - - 

one third of it by---- 

one fourth of it by---- 

and by the conditions, ■7r + -^H h45 = 448. 

-^ o 4 

Transposing - - -|- -f- -1 + -|. = 448 — 45 = 403 ; 

.-clearing of fractions, - - - - 6a; -f 4a: + 3a; = 4836 ; 

reducing, 13a; = 4836 ; 

lience, x=z 372. 

Let VIS see if this value will verify the equation. We have, 

3'yo 3-^2 372 

"2^ + -3- + -^ + 45 = 186 4- 124 + 93 + 45 = 448. 



82 ELEMENTS OF ALGEBRA. [CIIAP. IV. 

2. What number is that whose third part exceeds Its fourtli 
by 16? 

Let ths required aumber be denoted by x. 

Then, — x ^vill denote the third part ; 

o 

aiid -T-a; will denote the fourth part. 

4 

B}- the conditions of the problem, 

-^x--x= 16. 

Clearing of fractions, - 4iX — 3.t = 192 ; 
reducing, x =z 192. 

Verification. 

192 192 ,^ 

-3 r = ^^' 

or, - - - 16 = 16. 

3. Out of a cask of wine which had leaked away a third part, 
21 gallons were afterward drawn, and the cask was then half 
full :. how much did it hold 1 

Suppose the cask to have held x gallons. 

X 

Then, - - - - -^ "^'ill denote what leaked away; 

X 

and . - . . __ -}- 21 will denote what leaked out and 
o 

also what was drawn out. 

By the conditions of the problem. 



Clearing of fractions, - 2:c -f 126 = ox ; 

reducing _ a: — _ 126 ; 

dividing by ■— 1 - . x = 126. 

Verification. 

— + 21 _ — , 
or, 63 = 63. 



I 



CHAP. IV.l EQUATIONS OF THE FIRST DEGREE. 83 

4. A fish was caught whose ta'i weighed dlb. ; his head weighed 
as much as his tail and half his body ; his body weighed as much 
us his head and tail together : what was the weight of the fish 1 

Let - - 2x denote the weight of the body ; 
then - - 9 h a; will denote weight of the head ; 
and since the body weighed as much as both head and tail, 
2^ = 9+ 9 + x 
or, - 2x — X = 18', whence, re = 18. 

Verification, 

2x18-18 = 18; or, 18 :i= 18. 

Hence, the body weighed 86/55 ; 

the head weighed - 27/65 ; 

the tail weighed '-- 9Z65 ; 

and the whole fish 72/^5. 

5. A person engaged a workman for 48 days. For each day 
that he labored he received 24 cents, and for each day that he 
was idle, he paid 12 cents for his board. At the end of the 48 
days the account was settled, when the laborer received 504 
cents. Required the number of working days, and the nufnber oj 
days he was idle. 

If these two numbers were known, by multiplying them re- 
spectively by 24 and 12, then subtracting the last product from 
the first, the result would be 504. Let us indicate these 
operations by means of algebraic signs. 

Let . . X denote the number of working days ; 

then 48 — x will denote the number of idle days ; 

24 X a: = the amount earned, and 
12 (43 — x) =: the amount paid for his board. 
Then, from the conditions, 

24a; — 12(48- x) = 504 
or, 24a; - 576 -~ 12a; = 504. 

reducing 36a; = 504 + 576 = 1080 

whence. x =: oO the working days, 

and, 48 — 30= 18 the idle days. 



84 ELEMENTS OF ALGEBRA. [CHAP. IV. 

Yerification, 

Thirty days' labor, at 24 cents a day 
amounts to 30 x 24 = 720 cts ; 

and 18 days' board, at 12 cents a day, 
amounts to 18 X 12 = 210 cts ; 

and the amount received, is their difference, 504 cts. 

The preceding is but a particular case of a general problem 
which may be enunciated as follows. 

A. person engaged a workman for n days. For each day 
that he labored, he was to receive a cents, and for each day 
that he was idle, he was to pay h cents for his board. At 
the end of the time agreed upon, he received c cents. Re- 
quired the number of working days, and the number of idle 
days. 

Let ~ . X denote the number of working days ; then, 
n — X will denote the number of idle days ; 

ax will denote the number of cents he received; and 
b {n — x) will denote the number he paid out. 
From the conditions of the problem, 
ax — 6 {ii — a:) =3 c. 
Performing the indicated operations, transposing and factoring, 
we find, 

{a -\- h) X = c •\- hn^ 

whence, x = — ~T~> ^^® number of working days ; and 
n — X = p— -, the number of idle days. 

If we make n = 48, a = 24, b = 12 and c = 504, we obtain, 
504 + 576 



36 



= 30 ; and 48 — a; = 18 ; as before found. 



0. A fox, pursued by a greyhound, has a start of 60 leaps. 
He makes 9 leaps while the greyhound makes but 6 ; but 3 
leaps of the greyhound are equivalent to 7 of the fox. How 
uuuiy leaps must the greyhound make to overtake the fox? 



I 



CHAP. IV. J EQUATIONS OF THE FIRST DEGREE. 85 

Let us take one of the fox leaps as the unit of distance*, 

then, 3 leaps of the greyhound being equal to 7 ]eaps of the 

7 
fox, one of the greyhound leaps will be equal to -— . 

o 

Let X denote the number of leaps the greyhound must make 

before overtaking the fox. 

Then, since the fox makes 9 leaps while the hound makes 0, 

9 3 

r '' T" 

will denote the number of leaps the fox makes in the same time. 

7 
-— X will denote the whole distance passed over by the hound ; 

o 

— X will denote the whole distance passed over by the fox. 
Then, from the conditions of the problem, 

Clearing of fractions, 14a; = 3G0 + 9x, 

transposing and reducing, 5x = 360, 
whence, x = 72 ; 

3 3 

and -—-x = -—- X 72 = 108, the nrnt^bcr of fox leaps. 

Verification. 

7 X 72 ^^ , 3 X 72 
__ = 60+-^-., 

or, - - . t 168 = 168. 

7. A can 'do a piece of work alone in 10 days, and B in 13 
days : in what time can they do it if they work tog^th^i: ? 

Denote the number of days by x, and the work to ^ ^nie 
ly 1. Then, in 

1 day A can do — of the work ; and in 
1 day B can do — of the work ; lien<^e, iii 

X 

X days A can do — of the frork ; and in 

X 

% dlv7s B can do — of the work : 



86 ELEMENTS OF ALGOEA. [CHAP. IV. 

Hence, hj the conditions of the question, 

- + --1- 
10^13" ' 

clearing of fractions, 13a: + 102; = 130 : 

hence, x =z 5^^, the number of days. 

8. Divide $1000 between A, B and C, so that A shall have 
$72 more than B, and C $100 more than A. 

A?is. A's share = $324, B's = $252, C's = $424. 

9. A and B play together at cards. A sits down with $84 _ 
and B with $48. Each loses and wins in turn, when it ap- 
pears that A has five times as much as B. How much did A 
win? Ans. $26. 

10. A person dying, leaves half of his property to his wife, 
one sixth to each of two daughters, one twelfth to a servant, 
und the remaining $600 to the poor : what was the amount 
of his property ? Ans. $7200. 

11. A father leaves his propeiM;y, amounting to $2520, to four 
sons, A, B, G, and D. C is to have $3G0, B as much as C 
and D together, and A twice as much as B less $1000 : how 
much do A, B and D receive ? 

Ans. A $760, B $880, D $520. 

12. An estate of $7500 is to be divided between a widow, two 
sons, and three dauf^ters, so that each son shall receive twice as 
much as each daughter, and the widow herself $500 more than 
all the children • what ' was her share, and what the share of 
each child ? .' r Widow's share, $4000. 

Ans. \ Each son, $1000. 

' Each daughter, $500. 

• 

13. A company of ISO persons consists of men, women and 
children. The men are 8 more in number thr.n the women, and 
die children 20 more than the men and women together : how 
many of each sort m the company ? 

A71S. 44 men, 36 wom.en, 1*00 children. 



CHAP. IV. J EQUATIONS OF THE FIRST DEGREE. 87 

14. A father divides |2000 among five sons, so that each elder 
should receive $40 more than his next younger brother : what is "* 
the share of the youngest? - Atis. $320. 

15. A purse of $2850 is to be divided among three persoiLS, 
A, B and C ; A's share is to >be tt^^^ -^'^ share, and C is to , 
have $300 more than A and B together : what is each one's "-^ 
share? Ans. A's $450, B's $825, C's $1575. ^ 

16. Two pedestrians start from the same point; the first steps 
twice as far as the second, but the second makes 5 steps while 

the first makes but one. At the end of a certain time they are -A 
'iOO feet apart. Noav, allowing each of the longer paces to be 3 
feet, how far will each have traveled 1 

Ans. 1st, 200 feet; 2d, 500. 

17. Two carpenters, 24 journeymen, and 8 apprentices, re- 
ceived at the end of a certain time $144. The carpenters 
received $1 per day, each journeyman half a dollar, and each 
apprentice 25 cents : how many days were they employed ? 

Ans. 9 days. 

18. A capitalist receives a yearly income of $2940 • four fifths 

of his money bears an interest of 4 per cent., and the remainder / 
K)f five per cent. : how muoh has he at interest ? 

Ans. $70000^ 

19. A cistern containing 60 gallons of water has three unequal 
cocks for discharging it ; the largest will empty it in one hour, )/ 
Ihe second in two hours, and the third in three : in what tiiire 
will the cistern be emptiefJ if they all run together ? 

Ans. 32 J J min, 

20. In a certain orchard ^ are apple-trees, J peach-trees, 

I plum-trees, 120 cherry-trees, and 80 pear-trees : how many ^ 
trees in the orchard ? Ans. 2400. * 

21. A farmer being asked how many sheep he had, answered 
that he had them in five fields/ in the 1st he htid J, in the 
2d I, in the 3d |, in the 4th ^j, and in.the .^th 450 :' how 
many had he? Ans. 1200. 



88 ELEMENTS OF ALGEBRA. LCHAP. IV. 

^ 22. My horse and saddle together are worth $132, and the 
horse is worth ten times as much as the saddle : what is the 
value of the horse 1 Ans. $120. 

s^ 23. Tlie rent of an estate is this year 8 per cent, greater than 
it was last. This year it is |1890 : what was it last year 1 

Ans. $1750. 

24. What number is that from which, if 5 be subtracted, § of 
>^ the remainder will be 40 1 Ans. 65. • 

25. A post is i in the mud, J in the water, and ten feet above 
the water : what is the whole length of the post ? 

Ans. 24 feet. 

26. After paying J and J of my money, I had QQ guineas left 
iu my purse : how many guineas were in it at first ] 

Ans. 120. 

27 A person was desirous of givmg 3 pence apiece to some 
beggars, but found he had not money enough in his pocket by H 
pence ; he therefore gave them each two pence and had 3 pence 
remaining: required the number of beggars. Ans. 11. 

28. A person in play lost ^ of his money, and then won 3 
sliillmgs ; after which he lost -J of what he then had ; and this 
done, found that he had but 12 shillings remaining : what had 
he at first ? Ans. 20s. 

29. Two persons, A and B, lay out equal sums of money in 
trade ; A gains $126, and B loses $87, and A's money is new 
double B's : what did each lay out 1 Ans. $300. 

30. A person goes to a tavern with a certain sum of money 
in his phcket, where he spends 2 shillings; he then borrows 
as much money as he had left, and going to another tavern, 
he there spends 2 shillings also ; then borrowing again as 
much money as was left, he went to a third tavern, where, 
likewise, he spent 2 shillings and borrowed as much as he 
had left ; and again spending 2 shillings at a fourth taveru, 
he then had jotliing remaining;. What had he at first? 

Ans. Ss. 9d. 



CHAP. III.] EQUATIONS OF THE FIRST DEGREE. 80 

31. A farmer bought a basket of eggs, and offered them at 7 
cents a dozen. But before he sold any, 5 dozen were broken 
by a careless boy, for which he was paid. He then sold the re- 
mainder at 8 cents a dozen, and received as much as he would 
have got for the whole at the first price. How many eggs had 
lie in his basket? Ans. 40 dozen. 

Equations of the First Degree involving more than one 
Unknown Quantity. 

82* If we have an equation between two unknown quantities, 

we may find an expression for one of them in terms of the 

other and known quantities ; but the value of this unknown 

quantity could only be determined by assuming a value for 

the secoixd. Thus, from the equation, 

^ + 2y = 4, 
we may deduce 

a: = 4 -2y, - 

but cannot find a value for x without assuming one for ?/. 

If, however, we have another equation between the two un 
known quantities, the values of these quantities being the same 
in both, we may find, as before, an expression for x in terms 
of y, and this expression placed equal to the one already 
found, will give an equation containing but one unknown quan- 
tity. Let us take 

from which we find 

ic = 5 — oy. 

If we place this expression equal to that before found, we 

deduce the equation 

4 - 2y = 5 - 3y, 
from the solution of which we find, y = 1. 

This value of y, substituted in either of the given equations, 
gives X =z2: hence, 

re = 2 and y = 1 sadsfy both equations. 

We see that in order to find determinate values for two 
unknown quantities, we must have two independent equations. 
Simultaneous equations are those in which the values of the 
unknown quantities are the same in tlieni all at Me same time 



so ELEMENTS 3F ALGEBRA. [CHAP. IV. 

In the same manner it maj be sho\Yn that tc determine the 
values of three unknown quantities, we must have three equa- 
tions ; and generally, to determine the values of n unknown 
quantities we must have n equations. 

Elimination. 

83» JEUmination is the operation of combining several equationn 
■involving several unknown quantities^ and deducing therefrom, a less 
number of equations involving a less number of unknown quantities. 

Tliere are three principal methods of elimmation : 

bit. Bj addition or subtraction. 

2d. By substitution. 

Sd. By comparison. 

We shall explain these methods separately. 

Elimination by Addition or Subtraction, 

8-l» Let us take the two equations 
4a? — 5y = 5, 
Zx-\-2y^ 21. 
If we multiply both members of the first equation by 2, 
the co-efficient of y in the second, and both members of the 
second equation by 5, the co-efficient of y m the first, vre obtain, 
8^-10y= 10, 
Ibx + lOy = 105 ; 
in which the co-efficients of y are numerically the same in both. 
If, now, we add these equations member to member, we find 

23.r = 115. 
J[n this case y has been eliminated by addition. 
Again, let us take the equations 

2x^Zy— 12, 
Zx + 4.y= 17. 
If we multiply both members of the first equation by 8, 
the co-efficient of x in the second, and multn3ly both mem- 
bers of the second equation by 2, the co-efficient of x in the 

first, we shall have, 

6.r -h 9y = 36, 

6a; -f 8y = 34 ; 



CHAP. IV.] EQUATIONS OF THE FIKST DEGREE. 91 

in which the co-efficients of x are the same in both. If, now, 
vre subtract the second equation from the first, member from 
member, we find, 

2/ = 2. 

Here, x has been eliminated hj subtraction. 

In a similar manner we may eliminate one unknown quantity? 
between any two equations of the first degree containing any 
number of unknown quantities. Tiie rule for elimination by 
addition and subtraction may be simplified by using the least 
common multiple. Hence, for elimination by addition or sub- 
traction, we have the following 

RULE. 

Prepare the two equations in such a manner that the co -efficients 
of the qimtitity we wish to eliminate shall he numerically equal 
in both : then, if the two co-efficients have contrary signs, add the 
equations^ member to member ; if they have the same sign, sub- 
tract them member from member, and the resulting equation will 
he independent of that quantity. 

JElimination by Substitution, 

85. Let us take the equations, 

bx + ly = 4:Z, and 11a; -f % ^- 69. 
Find, from the first equation, the value of x in terms of y, 
wliich is, 

43 -7y 

Substitute this value for x in the second equation, and we 
shall have 

11 x(4.3-7y) 



+ 9y = 69: or 



reducing, ... 473 — 77^^ + 45?/ = 34.5. 

In a similar manner we may eliminate one unknown quantity 
between two equations of the first degree containing any number 
of unknown quantities. 

Hence, for eliminating by substitutior, we have the followins: 



92 ELEMENTS OF ALGEBRA. [CHAP. IV. 

♦ 

Find from one equation the value of the unJcnown quantity to 
be eliminated in terms of the others: suhstitute this value in ih$ |{ 
other equation for the unknown quantity to be eliminated, and the 
resulting equation will be independent of that quantity. 

Elimination hy Comparison. 



86» Let us take the equations, 




5a: 4- 7y = 43, 




11a; 4- 9y = 69. 




Finding the value of x in terms of y, 


from both equations 


we have, 




43 -7y 




_ 69 - 9y 





11 

If, now, we place these values equal to each other, we shall have, 
43 - 7y _ 69 - 9y 

5 "" n~' 

mlucing, ... 473 - 77y = 345 - 45y. 

Here, x has been eliminated. Generally, if we have two 
equations of the first degree containing any number of unknown 
quantities, any one of them may be eliminated by the following 

RULE. 

Find the value of the quantity we wish to eliminate, in terms 
of the others, from each equation, and then place these values 
equal to each other: the resulting equation will be independent 
of the quantity whose values were found. 

Tlie new equations which arise, from the two last method? 
of elimination, contain fractional terms. This inconvenience is 
avoided in the first method. The method by substitution is, 
however, advantageously employed whenever the co-efficient of 
either of the unknown quantities in one of the equatioas is equal 
to 1, because then the inconvanirnce of which we have just 



y ^J>1 4-10 - /SX- 

CHAP. IV.J EQUATIO:srS OP THE FIRST DEGREE. 9*3*5'*^ 

spoken doe: not o3Ciir. We shall sometimes have occasion to 
employ this method, but generally the method by addition and 
subtraction is preferable. When the co-efficients are not too 
great, the addition or subtraction may be performed at the 
same time with the multiplication that is made to render the 
oo-efficients of the same unknown quantity equal to each other. 

ITiore is also a method of elimination by means of thtt 
greatest common divisor, which will be explained in its appro»- 
priate place. 

87 • Let us now consider the case of three equations involving 
three unknown quantities. 

r 5x — Gt/ -{- 4z = 15, 

Take the equations, <7x-i-4ft/ — oz = 19, 
i2x-\- 2/ + 6.2 = 46. 

To eliminate z from the first two equations, multiply the first 
equation by 3 and the second by 4 ; and since the co-efficients 
of z have contrary signs, add the two results together : this gives 
a new equation, .... 43a; — 2y = 121."^ 

Multiplying both members of the second 
equation by 2, a factor of the co-efficient of 
z in the third equation, and adding them, 
member to member, we have - - - IQx -{- 9y =: 84.^ 

The question is then reduced to finding the values of x and y, 
which will satisfy these new equations. 

Now, if the first be multiplied by 9, the second by 2, and 
the results be added together, we find 

419a; = 1257, whence x = S. 

By means of the two equations involving x and y, we may 
determine y as we have determined x ; but the value of y may 
DC determined more simply, since by substituting for x ita 
value found above, the last of the two equations becomes, 
48 + 9y = 84, whence y = 4. 

In the sa^e manner, by substituting the values of x and y, 
the first of/ the three proposed equations becomes, 
15 - 24 + 4^ = 15, whence z = Q. 



94 ELEMENTS OF AUtiiBRA. [CHAP. lY. 

If we have a group of m simultaneous equations ccntaining m 
unknown quantities, it is evident, from principles already ex- 
plained, that the values of these unknown quantities may be 
found by the following 

RULE. 

I. Combine one of the m equations with each of the m — \ others, 
separately^ eliminating the same unknown quantity ; there loill result 
m — \ equations containing m — 1 unknown quantities, 

II. Combine one of these with each of the m — 2 others, sepa- 
rately, eliminating a second unknown quantity; there will result 
m — 2 equations containing m, — 2 unknown quantities. 

HI. Continue this operation of combination and elimination till 
we obtain, finally, one equation containing one unknown quantity. 

IV. Find the value of this unknown quantity by the rule for 
solving equations of the first degree containing one unknown quan- 
tity : substitute this value in either of the two preceding equations 
containing two unknown quonti-ties, and determine the value of a 
second unknown quantity : substitute these two values in either of 
the three equations involving three unknown quantities, and so on 
till we find the values of them all. 

It often happens that some of the proposed equations do nc 
contain all the unknown quantities. In this case, with a liuk 
address, the elimination is very quickly performed. 

Take the four equations involving four unknown quantities, 

2;p_Sy + 22r= 13 - (1) 4y + 22r=:]4 - (3). 

4M-2a:r=30 - (2) 5?/-|-3<^=:32 - (4). 

By examining these equations, we see that the elimination of 
z in equations (1) and (3), will give an equation involving x 
and y ; and if we eliminate u in the equations (2) and (4), we 
shall obtain a second equation, involving x and y. In the first 
place, the elimination of 2, in (1) and (3) gives ly — 2x =l \ - (5), 
that of u, in (2) and (4), gives - - 20y +\xc = 38 - (0). 

From (5) and (0) we readily deduce the values \ f y =iz 1 and 
x — o] and by substitution in (2) and (3^, we also\find u =9 
uiid 2 = 5. / 

/ 



^ 



CHAP. IV.J EQUATIONS OF THE FIRST DEGREE. 95 

EXAMPLES. 

1. Given 2x f 3y = IG, and 3j; — 2y == 1 1 to find the values 
of X and y, ^his. x = 5, y ~2. 

2. Given _ + --=_, and - + -- =^^ to find the 



values of x and y. 



^M5. oj — 



i) ' 



3. Given — -f- 7y = 99, and -^ -f 7i; = 51 to find the values 



of X and y. 

4. Given ^_ 12 - -f + §, and ~±^^ + ^ - 
2 4 5 

to find the values of x and y. 



^/i.9. a: — 7, y = 14. 
2;/ — X 

Ans. a; = 60, y =. 40. 



27 



5. Given 



r- ■ a: + y + = 29 >, 
^ fl;T|-'2y + 32 = 62 



to find X, y, and 2. 



6. Given 



7 Given 



8 Given 



Ans, X = S, y = 9, z =: 12. 

2x-h Ay — 35: = 22 j 

4.r — 2y + 5^ = 18 >■ to hnd .r, y, and z. 

6a; 4- 7y - ^ = 63 3 

^ri5. a^ =: 3, y = 1^ z = A. 

x-\-^y + ~z = S2 
y^4--^y + y2 = 15 [^ to find a;, y, and z. 

Ans. a: = 12, y = 20, 2 = 30. 
r 7a:^— 20+ 3w = 17 "^ 
4y- 22+ ^= ll^S 
5y - 3.r - 2?/ = 8 
4y — 3w + 2(! = /' 
32 + 8w = 33 j 
^/<s. .c = 2, y-4, 2 = 3, « = 3, < = 1. 



to find a:, y, z, m, 
and ^. 



96 ELEMENTS OF ALGEBRA. [CHAP. FV. 

PROBLEMS GIVING RISE TO SIMULTANEOUS EQUATIONS OF THE FIRfll 

DEGREE. 

1. What fraction is that, to the numerator of which, if 1 be 
ftdded, its value will be one third, but if 1 be added to its 
ienominator, its value widl be one fourth] 
Let X denote the numerator, and 

y the denominator. 
From the conditions of the problem, 
a; + l _ 2. 
y " 3' 

X 1 



y + 1 4 

Clearing of fractions, the first equation gives, 
3^: + 3 = y, 
and the 2d, 4a; = y + 1. 

Whence, by eliminating y, 

a: — 3 = 1, 
and a: = 4. 

Substituting, we find, 

2/ = 15; 

4 

and the required fraction is ■—:. 
^ Id 

2. To find two numbers such that their sum shall be equal 
to a and their difference equal to h. 
Let X denote the greater number, and 

y the lesser number. 
From the conditions of the problem, 
a; -h y = a, 
x — y — h. 
, Eliminating y by addition, 

2a; = a -f ft, 
a h 

""■ " = Y + T- 

By substitution, 

^~ 2 2' 



CHAP. IV.J EQUATIONS OF THE FIRST DEGREE ,07 

3. A person possessed a capital of 30000 dollars, for which 
}ie drew a certain interest per annum ; but he owed the sum 
of 20000 dollars, for which he paid a certain interest. The 
interest; that he received exceeded that which he paid by 800 
dollars. Another person possessed 35000 dollars, for which 
he received interest at the second of the above rates ; but he 
owed 24000 dollars, for which he paid interest at the first 
of the above rates. The interest that he received exceeded 
that which he paid by SIO dollars. Required the two rates 
of interest. 

Let X denote the first rate, and 

y the second rate. 

Then, the interest on $30000 at x per cent, for one year will be 

|30600a; ^^^^ 

-Jqq- or $300a:. 

The interest on $20000 at y per cent, for one year will be 

$20000y ^^^^ 

-j^ or $200y. 

Honce, Irom the first condition of the problem, 
300a; — WOy = 800 ; 
or, - - - . 3:r— 2y= 8 - - - (1). 
In like manner from the second condition of the problem we find 

35y- 24a; = 31 - - - (2). 
Combining equations (1) and (2) we find, 
y = 5 and x = 6. 

Hence, the first rate is 6 per cent, and the second rate 5 
per cent. 

Verification. 

f!30000, placed at 6 per cent., gives $300 X 6 = $1800. 

$20000 do 5 do $200 X 5 = $1000. 

And we have 1800 — 1000 = 800. 

The second condition can be verified in the same manner. 

4. There are three ingots formed by mixing together three 
metals in different proportions. 

7 



98 ELEMENTS OF ALGEBRA. [CHAP. IV 

One pound of the first contains 7 ounces of silver, 3 ounces 
of copper, and 6 ounces of pewter. 

One pound of the second contains 12 ounces of iilver. 3 ounces 
of copper, and 1 ounce of pewter. 

One pound of the third contains 4 ounces of silver, 7 ounces 
of copper, and 5 ounces of pewter. 

It is reqiired to form from these three, 1 pound of a fourth 
ingot which shall contain 8 ounces of silver, 3|- ounces of cop- 
per, and 4i ounces of pewter. 

}jet X denote the number of ounces taken from the first. 
y denote the number of ounces taken from the second 
z denote the number of ounces taken from the third. 

Now, since 1 pound or 16 ounces of the first ingot contains 7 

omices of silver, one ounce will contain 777 of 7 ounces : tiiat 

lo 



^' 16 


ounces ; and 
















X ounces will contain 


Ix 
16 


ounces 


of silver. 








y ounces will 


contain 


12j/ 
16 


ounces 


of silver, 








z ounces M-ill 


contain 


4z 
16 


ounces 


of silver. 






But 


since 1 pound of the new ingot is to 


contain 8 


ounces 


of 


silver. 


we have 

Ix 

16 


^ 16 ^ 16 


= 8; 









or, clearing of fractions, we have, 
for the silver, 7a: + 12y + 4z = 128 ; 

for the copper, 3a; + Sy + 7z = 60 ; 

and for the pewter, 6x -{■ y -{- 6z z= 68. 

Whence, finding the values of a:, y and 2, we have 

re = 8, the number of ounces taken from the first. 

y = 5 " " " " " " second. 

2 = 3 " " ,, " " " " tliird. 

5. What two numbers are ^-hey, whose sum is 33 and whos« 
difference is 7? Ang. 20 and 13. 



> 



y^ 



^' 



CHAP. IV. J EQUATIONS OF THE FIRST DEGKEE. 99 

6. Divide the number 75 into two such parts, that three times 
the greater may exceed , seven times the less by 15. 

Ans. 54 and 21. 

7. In a mixture of wine and cider, J of the whole plus 25 

gallons was wine, and ^ part minus 5 gallons, was cider ; how 

many gallons were there of each ? 

Ans. 85 of wine, and 35 of cider. 
« . 

8. A bill of £120 was paid in guineas and moidores, and the 
number of pieces of both sorts that were used was just 100 ; if 
the guinea were estimated at 21s., and the moldore at 275., ho 
many were there of each ? Ans. 50 

9. Two travelers set out at the same time from London and 
York, whose distance apart is 150 miles; they travel toward 
each other; one of them goes 8 miles a day, and the other 
7; in what time will they meet? Ans. In 10 days., 

10. At a certain election, 375 persons voted for two candi 
dates, and' the candidate chosen had a majority of 91 ; how 
many voted for each 1 ^ 

Ans. 233 for one, and 142 for the other. 



.0^ 



11. A's age is double B's, and B's is triple C's, and the sum 
of all their ages is 140; what is the age of each? 

Ans. A's = 84, B's = 42, and C's = 14. 

12. A person bought a chaise, horse, and harness, for £60 ; 
the horse came to twice the price of the harness, and the chaise 
to twice the price of the horse and harness ; what did he give 
for each 1 / £13 Gs. 8c?. for the horse. 

Ans. -j £ 6 13s. Ad. for the harness. 
V £40 for the chaise. 

13. A person has two horses, and a saddle worth £50 ; now, 
if the saddle be put on the back of the first horse, it will make 
his value double that of the second; but if it be^put on the back 
of the second, it will make his value triple that of the first 
what is th€> value of feach horse 1 

Ans. One £30, and the other £40. 






1 



100 ELEMENTS OF ALGEBRA. /, , '" [CHAP. W 

14. Two persd as, A and B, have each the same income, A 
saves I of his yearly; but B, by sp.endang £50 per annum mere 
than A, at the end of 4 years finds himself £100 in debt; what 
is the income of each? Ans. £125. 

15. To divide the number 36 into three such parts, that J of 
the first, J of the second, and -^ of the third, may be all equal 
to each other. Ans. \Q^ 12, and 16. 

16. A footman agreed to serve his master for £8 a year and 
A livery, but was turned away at the end of 7 months, and re 
reived only £2 135. 4c?. and his livery ; what was its value 1 

-^j_ ^f A71S. £4 165. 

17. To divide the number 90 into four such parts, that if the 
first be increased by 2, the second diminished by 2, the third 
multiplied by 2, and the fourth divided by 2, the sum, difference, 
product, and quotient, so obtained, w^ill be all equal to each other. 

Ans. The parts are 18, 22, 10, and 40. 

18. The hour and minute hands of a clock are exactly together 
at 12 o'clock ; when are they next together ] 

Ans. 1 h. 5j^j min. 

19. A man and his wife usually drank out a cask of beer in 
12 days ; but when the man was from home, it lasted the woman 
80 days; how many days would the man be in drinking it 
alone 1 Ans. 20 days. 

20. If A and B together can perform a piece of work in 8 
days, A and C together in 9 days, and B and C in 10 days; 
how many days would it take each person to perform the same 
work alone 1 Ans. A 14|| days, B 17f f , and C 23^. 

21. A laborer can do a certain work expressed by a, in a time 
expressed by 6; a second laborer, the work c in a time d-, a 
third, the work c in a time /. Required the time it would take 
the three la"':)orers. workirg together, to perform the work g. 



Ajis. 



adf ■\- bcf + bde 



r 

CHAP. IV.J EQUATIONS OF THE I [EST LEGREE. 10) 

22. If 32 2)ounds of sea water contair 1 pound of salt, how 
much fresh water must be added to these 32 pounds, in order NJ 
that the quantity of salt contained in 32 pounds of the new mix- ' 
ture shall be reduced to 2 ounces, or | of a pound % 

Ans. 224 lbs. 

23. A number is expressed by Inree figures ; the sum of these 
figures is 11; the figure in the place of units is double that in • 
the place of hundreds; and when 297 is added to this number^ ' 
the sum obtained is expressed by the figures of this number re- 
versed. What is the number % Ans. 326. 

2'!. A person who possessed 100000 dollars, placed the greater 
part of it out at 5 per cent, interest, and the other part at 4 per , . 
cent. The interest which he received for the whole amounted 
to 4640' dollars. Required the two parts. 

>/ „ Ans. $64000 and 836000. 

25. A person possessed a certain capital, which he placed out 
at a certain interest. Another person possessed 10000 dollars 
more than the first, and putting out his capital 1 per cent, more 
advantageously, had an income greater by 800 dollars. A third, 
possessed 15000 dollars more than the first, and putting out his 
capital 2 per cent, more advantageously, had an income greater 
by 1500 dollars. Required the capitals, and the three rates of 
interest. 

Sums at interest, $30000, $40000, $45000. 

Rates of interest, 4 5 6 per cent. 

26. A cistern may be filled by three pipes. A, B, C. By 
the two first it can be filled in 70 minutes ; by the firsc and 
third it can be filled in 84 minutes ; and by the second and 
third in 140 minutes. What time will each pipe take to do 
it in ? What time will be require!, if the three pipes run 
together 1 

/A in 105 minutes'. 

Ans. •< B in 210 minuter. 

( C in 420 minutes. 
All will fill it in one hour. 



A A 



102 ELEMENTS OF ALGEBRA. [CHAP. IV 

27. A, has 3 purses, each containing a certain sum of money 
If $20 be taken out of the first and. put into the second, il 
vrill contain four times as much as remains in the first. If $60 
be taken from the second and put into the third, then this will 
contain IJ times as much as there remains in the second. Again, 
if $40 be taken from the third and put into the first, then 
the third will contain 2| times as much as the first. What 
were the contents of each purse ? /1st. $120. 

Ans. \ 2d. $380. 
(3d. $500. 

28. A banker has two kinds of money ; it takes a pieces of 
the first to make a crown, and h of the second to make the 
same sum. Some one offers him a crown for <: pieces. How 
many of each kind must the banker give him % 

An^. 1st kmd, -^ -^ ; 2d kind, -^ -^. 

a — b a — b 

29. Find what each of three persons. A, B, C, is worth, 
knowing, 1st, that what A is worth added to I times what B 
and C are worth, is equal to p ; 2d, that what B is worth 
added to m 'times what A and C are worth, is equal to q ; 
3d, that what C is worth added to n times what A and B are 
worth, is equal to r. 

If we denote by s what A, B, and C, are worth, we intro- 
duce an auxiliary quantity, and resolve the question in a very 
simple manner, 

30. Find the values of the estates of six persons. A, B, C, D, 
E, F, from the following conditions : 1st. The sum of the estates 
of A and B is equal to a \ thnt of C and D is equal to h ; and 
that of E and F is equal to c. 2d. The estate of A is worth ra 
times that of C ; the estate of I) is worth n times that of E, and 
the estate of F is worth p times that of B. 

This problem may be solved by means of a single equation, 
iiivoh'ing but one unknown quantity. 



CHAP. IV.l EQUATIONS CF THE FIRST DEGREE. 168 

Of Indeterminate Equations and Indeterminate Problems. 

88» An equation is said to be indeterminate -^hen it may be 
satisfied for an infinite number of sets of values of the unknown 
quantities which enter it. 

Every single equation containing two unknown quantities is inde- 
terminate. 

For example, let us take the equation 
5x — S7j = 12, 

12 4- 3y 
whence, - - x = ~ . 

D 

Now, by making successively, t 

y=l, 2, 3, 4, 5, 6, &c., 

18 21 24 27 - . 

^ = 3, -, -, -, -, 6, &c., 

and any two corresponding values of x, y, being substituted in 
the given equation, 

5^ — 3y =r 12, 
will satisfy it: hence, there are an infinite number of values for 
X and y which will satisfy the equation, and consequently it is 
indeterrninate; that is, it admits of an infmite number of solutions. 
If an equation contains more than two unknown quantities, we 
may find an expression for one of them in terms of the others. 
If, then, we assume values at pleasure for these others, we 
can find from this equation the corresponding values of the first ; 
and the assumed and deduced values, t-iken together, will satisfy 
the given equation. Hence, 

Every equation involving more tnan one unknown quantity is 
indeterminate. 

In general, if we have n equations invohing more than n 
unknown quantities, these equations are indeterminate ; for we 
may, by combination and elimination, reduce them to a single 
equation ontaining more than one unknown quantity^ which we 
have already seen is indeterminate. 

If, on the contrary, we have a greater number of equations 
than we have unknown quantities, thoy cannot all be satisfied 



104: ELEMENTS OF A.LGEBRA. [CHA.P. IV. 

unless some of them are dependent upon the others. If we 
combine them, we may eliminate all the unlinoA\ai quantities, and 
the resulting equations, which will then contain only known 
quantities, will be so many equations of condition^ which must be 
f satisfied in order that the given equations n.ay admit of solution. 

Tor example, if we have 

X -\- y =a, 
x — y — c, 
xy = d; 

■we may combine the first two, and find, 

a c ^ a c 

" = 2 + 2 ^"^ 2'=2-25 
und by substituting these in the third, we shall find 
a^ c^ 

which expresses the relation between a, c and d, that must exist, 
in order that the three equations may be simultaneous. 

88*. A Prohlem is indeterminate when it admits of an iifinite 
number of solutions. This will always be the case wh^n its 
enunciation involves more unknow^n quantities than the;e are 
given conditions ; since, in that case, the statement of the p:'oblem. 
will give rise to a less number of equations than the.-e are 
unknown quantities. 

1st. Let it be required to find two numbers such that 5 
times the first dimirJshed by 3 times the second s^iU be 
equal to ]2. 

If we denote the numbers by x and y, the condition j of the 
problem will give the equation 

5x -Sy =: 12, 
which we have seen is indeterminate : — Hence, the ], ollem 
admits of an infinite number of solutions, or is indete" r -'nii-te. 

2. Find a quantity such that if it be multi23lied by a and 
the product increased by b, the result will be equal to "■ tim«» 
the quantity increased by d. 



CHAP. IV.] EQUATIONS OF THE FIRST DEGREE. 105 

Let, X denote the required qiiantit/. Then from the condiiion, 
ax ■\- h ■=. ex -{■ d^ 

whence, - - . a; = . 

a — c 

If now we make the suppositions that dz=.h and a — c, the 

voJue of X becomes -, which is a symbol o5 indeternr*ination. 

If we make these substitutions in the first equatio*^, it be 
comes 

ax -\- h :=^ ax -\- b, 
an identical equation (Art. 75), which must be satisfied for all 
values of x. These suppositions also render the conditions of 
the problem so dependent upon each other, that any quantity 
whatever will fulfil them all. 

Hence, the result - indicates that the problem admits of an 

infinite number of solutions. 

3. Find two quantities such that a times the first increased 
by b times the second shall be equal to c, and that d times 
the first increased by / times the second &hall be equal to (/. 

If we denote the quantities by x and y, we shall have from 
the conditions of the problem, 
ax -\- bi/ =z c, 

dx^fy^cj^ ' 

. cd — ag _ 

whence - y = -r-, ^, and 

bd — af 

If now we make 

cd = ay, (3) and af=z bd, (4) 
we shall find by multiplying these equations together, membei 
by member, 

cf= by. 

These suppositions, reduce the values of both x and v to — , 

Fiom (3) we find, 

d = "-l, an(3frc:n(4) f=±xd=^A 




108 ELEMENTS OF ALGEBRA. [CHAP. IV. 

whicli u' bstlt ited in equation (2), reduce it to 

ax -rhy = c, 
an eqvatior which is the same as the first. 

Undsr tliio supposition, we have in reality but one equal icn 
between two unknown quantities, both of which ought to be inde- 
terminate. This supposition also renders the conditions of the 
problem so dependent upon each other, as to produce a less 
number of ia dependent equations than there are unknown quan- 
tities. 

Generally, the result — , with the exception of the case men- 
tioned in Art. 71, arises from some supposition made upon the 
quantities entering a problem, which makes one or more condi- 
tions so dependent iTpon the others as to give rise to one or 

more indeterminate equations. In these cases the result -- is 

a true answer to the proKsm, and is to be interpreted as 
indicating that the problem admits of an infinite number of 
solutions. 

Interpretation of K^gative Results. 

89. From the nature of the signs -f and — , it is clear that 
the operations which they indica';e are diametrically opposite to 
each other, and it is reasonable to infer that if a positive re- 
sult, that is, one affected by the sign -f, is to be interpreted 
'h a certain sense, that a negative lesult, or one affected by 
whe sign — , should be interpreted in exactly the contrary 
sense. 

To show that tliis infer tnce is correct, we shall discuss one 
or two problems giving rise to both positive and negative 
results. 

1. To find a number, which added to the mnnbei- 6, will 
give a sum equal to the number a. 

Let X denote the required number. Then from iho i.-oudilions 
X -\-b =a^ v hence, x -z^a — h. 



CHAP. IV.] EQUATIONS OF THE FIRST DEGREE. 107 

This formv.la will give the algebraic value of a; in all the 
particular cases of the problem. 

For example, let a = 47 and 5 = 29 ; 
then, a; = 47 - 29 = 18. 

Again, let a = 24 and 5 = 31 ; 

then, ic = 24 - 31 uz - 7. 

This last value of a:, is called a negative solution. How is it 
to be interpreted? 

If we consider it as a purely arithmetical result, that is, as 
arising from a series of operations in which all the quantities 
are regarded as positive, and in which the terms add and suh- 
tract imply, respectively, augmentation and diminution, the prob- 
lem will obviously be impossible for the last values attributed 
to a and b ; for, the number b is already greater than 24. 

Considered, however, algebraically, it is not so ; for we have 
found the value of a; to be — 7, and this number added, in the 
algebraic sense, to 31, gives 24 for the algebraic sum, and there- 
fore satisfies both the equation and enunciation. 

2. A father has lived a number of years expressed by a ; his 
son a number of years expressed by b. find in how many years 
the age of the son will be one fourth the age of the father. 

Let X denote the required number of years. 

Then, a -\- x will denote the age of the father | at the end of the 

and b -\- X will denote the age of the son ) required time. 

Hence, from the conditions, 

a-\- X a — Ah 

— - — — -\- x\ whence, x = — - — . 

4 o 

-, . ^ 1 54-36 18 ^ 

Suppose a = 54, and 6 = 9; then x = =:-—•-= »5. 

o o 

The father being 54 years old, and the son 9, in 6 years tlie 
fu.her will be CO years old, and his son 15; now 15 is the 
f< urth of 60 ; hence, x = G satisfies the enunciation. 

IjCt us now suppo'.^5 a = 45, and i =r 15 ; 

45 - 60 

then, r = ^ = — 5, 

o 



108 ELEMENTS OF ALGEBRA. [CHAP. IV. 

f a; in 



If we substitute tliis value of a; in the eqjation, 
a -\- X 



we obtaiu, -^ — = 15 — 5 ; 



4 
45 — 5 
T 
or, 10 = 10. 

Hence, — 5 substituted for x^ verifies the equation, and there- 
fore is a true answer. 

Now, the positive result which was obtained, shows that the 
age of the father will be four times that of the son at the 
expiration of 6 years from the time when their ages were 
considered ; while the negative result, indicates that the age of 
the father was four times that of liis son, 5 years previous to 
the time when their ages were compared. 

The question, taken in its general, or algebraic sense, demands 
the time, when the age of the father is four times that of the 
son. In stating it, we supposed that the time was yet to 
come ; and so it was by the first supposition. But the con- 
ditions imposed by the second supposition, required that the 
time should have already passed, and the algebraic result con- 
formed to this condition, by appearing with a negative sign. 

Had we wished the result, under the second supposition, to 
have a positive sign, we might have altered the enunciation 
by demanding, how many years since the age of the father icas 
four times that of the son. 

If X denote the number of years, we shall ha re from the 

conditions, 

a — X . . 46 — a 

b — x\ hence, x =z — - — . 



4 

If a := 45 and b — 16, x will be equal to 5. 

From a careful consideration of the precedmg discussion, we 
may deduce the following principles with regard to negative 
results. 

1st. Every negative value found for the unknown qii%ntiiy from 
an equation of the first degree, will, when taken with **^ proper 
sign, satisfy the equation from which it was derived. 



CHAP IV.] EQUATIONS OF THE FIRST DEGREE. 109 

2d. This nerjative value, taken with its proper sign, will also 
satisfi the conditions of the problem, understood in its algebraic 
sense. 

o-d. If a positive result is interpreted in a certain sense, a nega^ 
tive result must be interpreted in a directly contrary sense. 

4th. The negative result^ with its sign changed, may be regarded 
as the answer to a problem of which the enunciation only differs 
from the one proposed in this : that certain quantities which were 
additive have become subtractive, and the reverse. 

90. As a further illustration of the extent and power of the 
algebraic language, let us resume the general problem of the 
laborer, already considered. 

Under the supposition that the laborer receives a sum c, we 

have the equations 

X -\- y = n) . bn + c an — c 

y whence, x — -, y = — — . 

ax — by = c ) a -{- b a -\- b 

If, at the end of the time, the laborer, instead of receiving 
a sum c, owed for his board a sum equal to c, then, by would 
be greater than ax, and under this supposition, we should have 
the equations, 

X -\- y :=^ n, and ax — by z= — c. 

Now, since the last two equations differ from the preceding 
two given equations only in the sign of c, if we change the 
sign of c, in the values of x and y, found from these equations, 
the results will be the values of x and y, in the last equa- 
tions : this gives 



bn — c 



an 



a + 6' ^ a-\-b' 

The results, for both enunciations, may be comprehended in 

tlie same formulas, by writing 

bndzc an zri c 
y — ' 



a-\-V " a + b' 

The double sign ±, is read plus or minus, and qp, is read, 

minus or plus. The upper signs correspond to the case in 

which the laborer received, and the lower signs, to the case iu 



110 ELEMENTS OF ALGEBRA. [CHAP. IV. 

which he owed a sum c. Tliese formulas also comprehend the 
case in which, in a settlement between the laborer and hid 
employer, their accounts balance. This supposes c =: 0, which 

gi 788 

bfi an 



Discussion of Proble'ms. 

91. Tlie discussion of a problem consists in mak.ng every 
possible supposition upon the arbitrary quantities whioh enter 
the equation of the problem, and interpreting vae results. 

An arbitrary/ quantity, is one to which we may assign a valvt^ 
at pleasure. 

In every general problem there is always one or more arbi 
trary quantities, and it is by assigning particular values to 
these that we get the particular cases of the general problem. 

The discussion of the following problem presents nearly all 
the circumstances which are met with in problems giving rise 
to equations of the first degree. 



PROBLEM OF THE COURIERS. 



ma 



Two couriers are traveling along the same right line an 
in the same direction from R' toward R. The number of miles 
traveled by one of them per hour is expressed by 7?z, and the 
number of miles traveled by the other per hour, is expressed 
by n. Now, at a given time,- say 12 o'clock, the distance be- 
tween them is equal to a number of miles expressed by a : re- 
quired the time when they are together. 

.. -^ R' A B ' R. 



At 12 o'clock, suppose the forward courier to be at B, the 
other at A, and R or R' to be the point at which they are 
together. 

Let a denote the distance AB, between the couriers at 13 
o'clock, and suppose that distances measured to the right, from 
A. are regarded as positive cpiantities. 



.,,^:>.s ^kc^^^i-cu.!/-^ 



CHAP. IV. J EQUATIONS OF 1?HE FIRST DEGREE. V HI 

Let t denote the number of hours fnm 12 o'clock to the 
time when they are together. 

Let X denote the distance traveled by the forward courier 
in t hours ; 

Then, a-\- x will denote the di stance trave' 'xi by the other ' 
in the same time. 

Now, since the rate per hour, multiplied by the number of 
hours, gives the distance passed over by each, we have, 
i X m n= a + a; - - - - (1) 
ty, n —X .... (2). 

Subtracting the second equation from the first, member from 
member, we have, / 4/ 

whence, - - - - ^ = . 

m — n 

We will now discuss the value of ^ ; a, m and n^ bein^ 
arbitrary quantities. > ^'*^ <; 

First^ let us suppose m y n. ^ 

The denominator in the value of ^, is then positive, and since 
a is a positive quantity, the value of t is also positive. 

lliis result is interpreted as indicating that the time when 
they are together is after 12 o'clock. 

Tlie conditions of the problem confirm this interpretation. 

For if m > w, the courier from A will travc4 faster than the 
courier fi'om B, and will therefore be contiuaally gaining on 
him : the interval which separates them will diiriinish more and 
more, until it becomes 0, and then the couriers will be found 
upon the same point of the line. 

In this case, the time ^, which elapses, must bti added to 12 
o'clock, to obtain the time when they are together. 

Secondi, suppose m < 71. 

The denominator, m — n will then be negative, and the value 
of / will also be negative. 

Tliis result is interpreted in a sense exactly contrary to the 
mterpretation of the positive result; that i.^, it indicates that 
the time of their being together was previous to 12 o'clock. 



112 ELEMENTS OF ALGEBRA. [CHAP. IV. 

This interpretation is also confirmed by considering the 
oircumstances of the problem. For, under the second suppo- 
sition, the courier which is in advance travels the fastest, and 
therefore will continue to separate himself from the other 
oourier. At 12 o'clock the distance between them was equal 
to a : after 12 o'clock it is greater than a ; and as the rate 
of ti-avel has not been changed, it follows that previous to 12 
o'clock the distance must have been less than a. At a certain 
honr, therefore, before 12, the distance between them must have 
b^en equal to nothing, or the couriers were together at some 
point R'. The precise hour is found by subtracting the value of 
t from 12 o'clock. -^ 

Third^ suppose m =z n. 

The denominator m — n will then become 0, and the value 

of t will reduce to -, or od . 

This result indicates that the length of time that must elapse 
before they are together is greater than any assignable time, or 
in other words, that they will never be together. 

This interpretation is also confirmed by the conditions of the 
p"r*oblem. 

For, at 12 o'clock they are separated by a distance a, and if 
m =n they must travel at the same rate, and we see, at once, 
that whatever time we allow, they can never come together j 
hence, the time that must elapse is infinite. 

Fourth] suppose a =: and m > n or m < n. 

The numerator being 0, the value of the fraction is oi 
/ = 0. 

This result indicates that they are together at 12 o'clock, 
or that there is no time to be added to or subtracted from 
12 o'clock. 

The conditions of the problem confirm this interpretation. 
Because, if a = 0, the couriers are together at 12 o'clock ; and 
since they travel at different rates, they could never have been 
together, nor can they be together after 12 o'clock : hence, t can 
have no other value than 0. 



«mA.P. IV.] OF INEQUALITIES. US 

Fifth, suppose a = and m =^n. 
The value of t becomes -, an indeterminate result. 

This indicates that t may have any value whatever, or in 
other words, that the couriers are together at any time either 
before or after 12 o'clock : asid this too is evident from the cir 
cum stances of the problem. 

For, if a = 0, the couriers are together at 12 o'clock ; and 
since they travel at the same rate, they will always be together; 
hence, t ought to be indeterminate. 

The distances traveled by the couriers in the time J are, 

respectively, 

ma ^ na 
and - 



m — n m — n! 



both of which will be plus when m'^n, both minus whenm < n^ 
and infinite when m =z n. 

In the first case t is positive ; in the second, negative ; and in 
the third, infinite. 

When the couriers are together before 12 o'clock, the distances 
are negative, as they should be, since we have agreed to call 
distances estimated to the right positive, and from the rule for 
interpreting negative results, distances to the left ought to bo 
regarded as negative. 

Of Inequalities. ~ i /^ 

92» An inequality is the expression of two unequal quantities 
connected by the sign of inequality. 

Thus, a > & is an inequality, expressing that the quantity a 
is greater than the quantity b. 

The part on the left of the sign of inequality is called the first 
member, that on the right the second member. 

The operations which may be performed upon equations, may 
m general be performed upon inequalities; but there are, never- 
theless, some exceptions. 

In order to be clearly understood, we will give examples of 
the diiferent transformations to which ir equalities may be suli 

8 



114 ELEMENTS OF ALGEBRA. [CHAr. IV. 

jected, taking care to point out the exceptions to which these 
transformations are liable. 

Two inequalities are said to subsist in the same sense, when 
the greater quantity is in the first member in both, or in the 
second member in both; and in a contrary sense, when the 
greater quantity is in the first member of one and in the second 
member of the other. 

Thus, 25 > 20 and 18 > 10, or 6 < 8 ar.d 7 < 9, 
are inequalities which subsist in the same sense ; and the in 
equalities 

15 > 13 and 12 < 11, 
subsist in a contrary sense. 

1. If we add the same quantity to both members of an inequality^ 
or subtract the same quantity from both members^ the resulting 
inequality will subsist in the same sense. 

Thus, take 8 > 6 ; by adding 5, we still have 
o -h 5 > 6 + 5 ; 
and subtracting 5, we have 

8 - 5 > 6 - 5. 

When the two members of an inequality are both negative, 
that one is the least, algebraically considered, which contains the 
greatest number of units. 

Thus, — 25 < — 20 ; and if 30 be added to both members, 
we have 5 < 10. This must be understood entirely in an alge- 
braic sense, and arises from the convention before established, to 
(consider all quantities preceded by the minus sign, as subtractive. 

The principle first enunciated serves to transpose certain terrns 
from one member of the inequality to the other. Take, for ex 
ample, the inequality 

«2+ 52->352_2a2; 

there will result, by transposing, 

a2 + 2a2 > 362 - 52^ oj. 3^2 > 2^2. 

2. If two inequalities subsist in the same sense, and we add fJutm 
member to member, the resulting inequality wiU also subsist in the 
same sense. -^-^ 



1 



CHAP. IV.J OF INEQUALITIES. 115 

Thus, if we add a > 6 and c^ d, member to member, 
there results a + c > 6 + c?. 

But this is not ahcays the case, when we subtract, member from 
member, two inequalities established in the same sense. 

Let there be two inequalities 4 < 7 and 2 < 3, we have 

4-2 or 2 < 7 - 3 or 4. 
But if we have the inequalities 9 < 10 and 6 < 8, by sub- 
tracting, we have 

9—6 or 3>10 — 8 or 2. 

We should then avoid this transformation as much as possible, 
or if we employ it, determine in which sense the resulting in- 
equality subsists. 

3. If the two members of an inequality be multiplied by a 
positive quantity, the resulting inequality will exist in the same 
sense. 

Thus, - - - o < 6, will give 3a < 35 ; 

and, - - - - — a < — b, — 3a < — 36. 

This principle serves to make the denominators disappear. 

T^ , . ,. a^ — b"^ <? — d^ 

t rom the mequality — — — > — , 

tta o(x 

we deduce, by multiplying by ^ad, 

3a(a2 - &2) y 2d (c2 — dP), 

and the same principle is true for division. But, 

When the two members of an inequality are multiplied or 
divided by a negative quantity^ the resulting inequality will sub- 
tlst in a contrary sense. 

Take, for example, 8 > 7 ; multiplying by — 3, we have 

- 24 < - 21. 

8 8 7 

In like manner, 8 > 7 gives — — , or — < — . 

— o 3 3 

Therefore, when the two members of an inequality are multi- 
plied or divided by a quantity, it is necessary to ascertain 
whether the multiplier or divisor is negative; for, in that case, 
the inequality will exist in a contrary sense. 



116 ELEMENTS OF ALGEBRA. [CHAP. IV 

4. It is v,ot permitted to change the signs of the two members 
of an inequality^ unless we establish the resulting inequality in a 
contrary sense; for, this transformation is evidently the same as 
multiplying the two members by — 1. 

5. Both members of an inequality between positive quantities 
ean he squared^ and the inequality will exist in the same sense. 

Thus, fi'om 5 > 3, we deduce, 25 > 9 ; from a -[- 6 > c, we 

find 

(a + by > c2. 

6. When the signs of both members of the inequality are not 
known, we cannot tell before the ojjeration is pej formed, in which 
sense the resulting inequality will exist. 

For example, — 2 < 3 gives (--2)2 or 4 < 9. 

But, 3 > — 5 gives, on the contrary, (3)^ or 9 < ( — 5)^ 
or 25. 

We must, then, before squaring, ascertain the signs of the two 
members. 

Let us apply these principles to the solution of the folloT\*ing 
examples. By the solution of an inequality is meant the oper 
ation of finding an inequality, one member of ^\^ich is the 
unknoy^'n quantity, and the other a known expression. 





EXAMPLES. 




\ 






1. 


5x-Q> 19. 




Ans. 


a:>5. 


2. 


3ar + ^a;-30>10. 






Ans. 


x>4. 


3. 
4. 


X 1 , or 13 ^ 17 

6 3 ^2^2-^2 


^ 




Ans. 
Ans. 


x>Q. 
x>a. 


5. 


y - at + oi < y. 






Ana. 


x<b. 



CHAPTER V. 

EXTRACTION OF THE SQUARE ROOT OF NUMBERS.-- ORMATION OP TH3 
SQUARE AND EXTRACTION OF THE SQUARE ROOT OF ALGEBRAIC QUANTI- 
TIES. TRANSFORMATION OF RADICALS OF THE SECOND DEGREE. 

93« The square or second power of a number, is the product 
which arises from multiplying that number by itself once : for 
example, 49 is the square of 7, and 144 is the square of 12. 

llie square root of a number, is that number which multiplied 
by itself once will produce the given number. Thus, 7 is the 
square root of 49, and 12 the square root of 144 : for, 7x7 = 49, 
and 12 X 12 = 144. 

The square of a number, either entire or fractional, is easily 
found, being always obtained by multiplying the number by itself 
once. The extraction of the square root is, however, attended 
with some difficulty, and requires particular explanation. 

The first ten numbers are, 

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 
and their squares, 

1, 4, 9, 16, 25, 36, 49, 64, 81, 100. 

Conversely, the numbers in the first line, are the square roots 
of the corresponding numbers in. the second line. 

We see that the square of any number, expressed by one 
f.gure, will contain no unit of a higher order than tens. 

The numbers in the second line are perfect squares^ and, 
generally, any number which results from multiplying a whole 
number by itself once, is a perfect square. 

If we wish to find the square root of any number less than 
100, we look in the second line, above given, and if the num- 
ber is there written, the corresponding number in he first line 



118 ELEMENTS OF ALGEBRA. fCHAP. V. 

is its square root. If the number falls between any two num 
bers in the second line, its square root will fall between the 
corresponding numbers in the first line. Thus, 55 falls between 
49 and 64 ; hence, its square root is greater than 7 and less 
than 8. Also, 91 falls between 81 and 100 ; hence, its square 
root is greater than 9 and less than 10. 

If now, we change the units of the first line, 1, 2, 3, 4, <fec.,, 
into units of the second order, or tens, by annexing to each, 
we shall have, 

10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 
and their corresponding squares will be, 

100, 400, 900, 1600, 2500, 3600, 4900, 6400, 8100, 10000: 
Hence, the square of any number of tens will contain no unit oj 
a less denomination than hundreds. 

94. We may regard every number as composed of the sun? 
of its tens and units, 

JMow, if we represent any number by i^, and denote the 
tens by a, and the units by b, we shall have, 

JSr=a-\-b; 
whence, by squaring both members, 

iV^2 _ ^2 _|_ 2ab + 62 : 

Hence, the square of a number is equal to the square of the 
tens, plus twice the product of the tens by the units, plus the sqiuire 
of the units. 

For example, 78 = 70 + 8, hence, 

(78)2 = (70)2 4. 2 X 70 X 8 + (8)2 =z 4900 + 1120 + 64 = 6084. 

95. Let us now find the square root of 6084. 
Since tliis number is expressed by more than two 

figures, its root will be expressed by more than one. 60 84 

But since it is less than 10000, which is the square 
of 100, the root will contain but two places of figures ; that 
is, i:nits and tens. 

Now, the square of the number of tens must be found in the 
number expressed by the two left-hand figures, which we will 
separate from the other two, by placing a point over the plac-e 



60 84 1 78 
49 

7 x2 =14 8 



1184 
1184 
■ 



CHAP, v.] SQUAEli BOOT OF NUMBERS. 119 

of units, and another over the place of hundreds. These parts, 
of two figures each, are called periods. The part 60 is com- 
prised between the tM'O squares 49 and 64, of which the roots 
are 7 and 8 : hence, 7 is the number of tens sought ; and the 
required root is composed of 7 tens plus a certain number of 
units. 

The number 7 being found, we 
set it on the right of the given 
number, from which we separate 
it by a vertical line : then we 
subtract its square 49 from 60, 
which leaves a remainder of 11, 
to which we bring down the two 
next figures 84. The result of this operation is 1184, and this 
number is made up of twice the product of the tens by the units 
plus the square of the units. 

But since tens multiplied by units cannot give a product of a 
lower order than tens, it follows that the last number 4 can 
form no part of double the product of the tens by the units : 
this double product, is, therefore found in the part 118. 

Now, if we double the i^umber of tens, which gives 14, and 
then divide 118 by 14, th . quotient 8 is the number of units of 
the root, or a greater number. This quotient can never be too 
small, since the part 118 will be at least equal to twice the 
product of the number of tens by the units: but it may be too 
large; for the 118, besides the double product of the number 
of tens by the units, may likewise contain tens arising from 
the square of the units. 

To ascertain if the quotient 8 expresses the number of units, 
we place the 8 to the right of the 14, which gives 148, and then 
we multiply 148 by 8 : Thus, we evidently form, 

1st, the square of the units ; and 

2d, the double product of the ;.ens by the units. 

TTiis multiplication being affected, gives for a product 1184, 
the same number as the result of the first operation. Having 



120 ELEMENTS OF ALGEBRA. [CHAP. V, 

subtracted the product, we find the remainder equal to : heiico 
78, is the root required. 

Indeed, in the operations, we have merely- subti acted from the 
given number 6084, 
' 1st, the square of 7 tens or of 70 ; 

2d, twice the product of 70 by 8 ; and 

3d, the square of 8: that is, the three parts which enter mt<i 
■ tiie composition of the square of 78. 

In the same manner we may extract the square root of any 
number expressed by four figures. 

95. Let us now extract the square root of a number expressed 
by more than four figures. 

Let 56821444 be the number. 56 82 14 44 | 7538 

If we consider the root as the 49 

sum of a certain number of tens 14 5 782 
and a certain number of units, the 725 

given number will, as before, be 150 3 57'J 4 

equal to the square of the tens plus 450 9 

twice the product of the tens by ISOdS 

the units plus the square of the units. 

If then, as before, we point off 
a period of two figures, at the right, the square of the tens of the 
required root will be found in the number 568214, at the left; 
and the square root of the greatest perfect square in this number 
"will express the tens of the root. 

But since this number, 568214, contains more than two figures, 
its root will contain more than one, (or hundreds), and the >qjare 
of the hundreds will be found in the figures 5682, at the left of 14 ; 
hence, if we poi»it off a second period 14, the square root of the 
greatest perfect square in 5682 will be the hundreds of the required 
root. But since 5682 contains more than two figures, its root will 
contain more than one, (or thousands), and the square of the thousands 
will be found in 56, at the left of 82 : hence, if we point off a third 
period 82, the square root of the greatest perfect square in 56 will 
be the thousands of the required root. Hence, we place a point 
over 56. and then nroceed thus : 



12054 4 
12054 4 



(IHAP. V.J SQUAKE ROOT OF NUMBERS. 121 

Placing 7 on the right of the given number, arxd subtracting 
its square, 49, from the left hand period, we find 7 for a remain- 
der, to which we annex the next period, 82. Sepai'ating the last 
figure at the right from the others by a point, and dividing the 
number at the left by twice 7, or 14, we have 5 for a quotient 
figure, which we place at the right of the figure already found, 
and also annex it to 14, Alultiplying 145 by 5, and subtracting 
the product from 782, we find the remainder 57. Hence, 75 is 
the number of tens of tens, or hundreds, of the required square 
root. 

To find the number of tens, bring down the next period and 
annex it to the second remainder, giving 5714, and divide 571 
by double 75, or by 150. The quotient 8 annexed to 75 gives 
753 for the number of tens in the root sought. 

We may, as before, find the number of units, which in this 
case will be 8. Therefore, the required square root is 7538. A 
similar course of reasoning may be applied to a number expressed 
by any number of figures. Hence, for the extraction of the 
square root of numbers, we have the following 

RULE. 

I. Separate the given number into periods of two Jigitres each, 
beginning at the right hand: the period on the left will often con- 
tain but one figure. 

II. Find the greatest perfect square in the first period on the 
left^ and place its root on the right afer the manner of a quotient 
in division. Subtract the square of this root from the first 
period, and to the remainder bring down the second period for a 
dividend. 

ill. Double the root already found and place it on the left for a 
divisor. See how many times the divisor is contained in the 
dividend^ exclusive of the right hand figure^ and place the quotient 
ill the root and also at the right of the divisor. 

iV. Multiply the divisor, thus augmented, by the last figure 
of the root found^ and subtract the product from the dividend 



122 ELEMENTS OF ALGEBRA. CHAP. V, 

and to the remcuy der bring iown the next period for a new 

dividend. 

V. Double the whole root already found, for a new divisor, 
and continue the operation as before, until all the periods are 
brought down. 

Remark I. — If, after all the periods are brought down, there 
is no remainder, the proposed number is a perfect square. But 
if there is a remainder, we liave only found the root of the 
greatest perfect square contained in the given number, or the 
entire part of the root sought. 

For example, if it were required to extract the square root of 
168, we should find 12 for the entire part of the root and a 
remainder of 24, which shows that 168 is not a perfect square. 
But is the square of 12 the greatest perfect square contained 
in 168 1 That is, is 12 the entire part of the root ? "^ 

To prove this, we will first show that, the difference between 
the squares of two consecutive numbers, is equal to twice the less 
number augmented by 1. 

Let a represent the less number, 

and a + 1, the greater. 

Tlien, (a-f-l)2 = a2 + 2a + l, 
and (a) 2 := a^, 



their difference is 2a + 1 as enunciated : hence, 

The entire part of the root cannot be augmented by 1, unless 
the remainder is equal to, or exceeds twice the root found, plus 1. 

But, 12 X 2 + 1 r= 25 ; and since the remainder 24 is less 
than 25, it follows laat 12 cannot be augmented by a number 
as great as unity : hence, it is the entire part of the root. 

The principle demonstrated above, may be readily applied in 
finding the squares of consecutive numbers. 

If the numbers are large, it will be much easier to apply the 
above pifnciple than to square the numbers separately. 



CHAP, v.] SQUAKE ROOT OF NUMBERS. 128 

For example, if we have (651)2 ^ 423801, 
and wish to find the square of 652, we have, 
(651)2 _ 423801 
+ 2 X 651 = 1302 

+ 1 = 1 

and 

Also, 





(652)2 


rz 425104. 




(652)2 


= 425104 


+ 2 


X652 


= 1304 




+ 1 


= 1 




(653)2 


= 426409. 



and 

Remark II. — The number of places of figures m the root 
will always be equal to the number of periods into which the 
given number is separated. 

EXAMPLES. 

1. Find the square root of 7225. 

2. Find the square root of 17689. ^ / ^ o 

3. Find the square root of 994009. ^ 9 f ) 

4. Find the square root of 85678973. - *^1 5'^ 

5. Find the square root of 67812675. ^ ^ . 

6. Find the square root of 2792401. - ; ^ y f ^;. 

7. Find the square root of 37496042. Z / / ; 5 

8. Find the square root of 3661097049. i,CJr) 

9. Find the square root of 918741672704.^ ^ j*^4 ^"^ ^ 

Remark III. — The square root of an imperfect square, is in 
' commensurable with 1, that is, its value cannot be expressed 
in exact parts of 1. 

To prove this, we shall first show that if -r- is an irreduci' 



ble fraction, its square —5 must also be an irreducible fraction. 

A number is said to be prime when it cannot be exactly di- 
vided by any other immber, except 1. Thus 3, 5 and 7 are 
prime numheri. 



J 



124 ELEMENTS OF ALGEBRA. [CHAr. V, 

It is a fundamental principle, that every number may be re- 
solved into prime factors, and that any number thus resolved, 
is equal to the continued product of all its prime factors. It 
often happens that some of ihes^, factors are equal to each 
other. For example, the number 

50 = 2 X 5 X 5 ; and, 180 =:= 2 X 2 x 3 X 3 x 5. 

Now, from the rules for multiplication, it is evident that the 
square of any number is equal to the continued product of all 
the prime factors of that number, each taken twice. Hence, we 
see that, the square of a number cannot contain any 'prime factor 
which is not contained in the number itself. 

But, since — , is, by hypothesis, an irreducible fraction, a 

and b can have no common factor : hence, it follows, from 
what has just been shown, that a^ and b" cannot have a com- 

0,2 

mon factor, that is, — is an irreducible fraction, which was 

to be proved. 

a^ tt* a. ** 

For like reasons, — -, t— , - - t— , are also irreducible fractions. 
b^ 6* 6" 

Now, let c represent any whole number which is an imper 

febt square. If the square root of c can be expressed by » 

fraction, we shall have 

- a 

in which -r- is an irreducible fraction. 
b 

Squaring both members, gives, 

a? 

'--¥^ 

or a whole number equal to an irreducible fraction, whicli is 
absurd ; hence, Vc~ cannot be expressed by a fraction. 

We conclude, therefore, that the square root of an imperfect 
square cannot be expressed in exact parts a 1. It may be 
'^hown, in a similar manner, that ony root :f an imioerfect 
power of the degree indicated^ cannot he exp-^essed in exact p)orti 

of 1- 



CHAP. V.} SQUARE ROOT OF FRACTIONS. 125 

Extraction of the Square Root of Fractions, 

96. Since the second power of a fraction is obtained by- 
squaring thfl numerator and denominator separately, it follows 
that the square root of a fraction will be equal to the square rout 
of the numerator divided by the square root of the denominator. 



For example, \/"l2 ~ 



a a a?" 

But if the numerator and the denominator are not both per- 
fect squares, the root of the fraction cannot be exactly found. 
We can, however, easily find the root to within less than the 
fractional unit. 

Thus, if we were required to extract the square root of the 

fraction --, to within less than — , multiply both terms of the 

fractions by 6, and we have — . 

Let r2 represent the greatest perfect square in aJ, then will 
ah be contained between r^ and (r + 1)^, and — will be con- 
tained between 

Jl and (" + ^)' 

and the true square root of — = — , vnll be contained b^ 

tween 

r r-f 1 

T r + 1 1 - 

but the difference between -r- and — - — is — ; hence, either 

b b 

will be the sq-jare root of -r-, to withm less than -j-. We have 

6 



then the folio wirg 



126 ELEMENTS OF ALGEBRA. ICHA.P. V. 

HULE. 

Multiply the numerator by the denominator ^ and extract th§\ 
square root of the product to within less than 1 ; divide i?i*\ 
result by the denominator, and the quotient will be the apprcxi- 
mate root. 

For example, to extract the square root of — , we multiply 

3 hj 5, which gives 15 ; the perfect square nearest 15, is 16, 

and its square root is 4 ; hence, — is the square root of --- 

to within less than — -. 
5 

97. If we wish to determine the square root of a whole 

number which is an imperfect square, to within less than a 

given fractional unit, as — , for example, we have only to plac« 

the number under a fractional form, having the given fractional 
unit (Art. 63), and then we may apply the preceding rule: or 
what is an equivalent operation, we may 

Multiply the given number by the square of the denominator 
of the fraction which determines the degree of approximation ; then 
extract the square root of the product to the nearest unit, and 
divide this root by the denominator of the fraction. 

r 
EXAMPLES. 

1. Let it be required to extract the square root of 59, to 
within less than — . 

First (12)2 = 144 ; and 144 x 59 = 8496. 

Now, the square root of 8496 to the nearest unit, is 92 : henc« 

92 1 

T-^ — T'y^Y? which is true to within less t'aan -— . 

2. Find the .^JIa to within less than — . Jns. 3^. 



S. Find the .^^223 to within less than — . Ans. 14fJ. 



15 
I 
40' 



CHAP, v.] SQUARE EOOT OF FRACTIONS. 127 

97*« The nxanner of determining the approximate root in deci- 
mals, is a consequence of thA ^preceding lule. 

/ '' '" J 1 

To obtain the square root -W an,, "entire number within — , 

1 1 /f^ 

— , , &c., it is only neccs^ify, according to the preceding 

rule, to multiply the proposed number by (10)^, (lOO)^, (1000)^; 
or, which is the same thing, 

Annex to the number^ two, four^ six, c()c., ciphers : then extract 
the root of the product to the nearest unit, and divide this root 
by 10, 100, 1000, &c., which is effected hg pointing off one, two, 
three, d'C, decimal places from the right hand. 

EXAMPLES. 

1. To fmd the square root of 7 to within less than ■—-. 

Having multiplied by (100)^, that is, 
having annexed four ciphers to the right 
hand of 7, it becomes 70000, whose 
root extracted to the nearest unit, is 264, 
which being divided by 100 gives 2.64 
for the answer, and this is true to within 



7 00 00 2.04 
4 



300 

276 



2400 
2096 



'ess than -^. 3O4 Rem. 

2. Find the ^29 'jo mthin less than ----. Ans. 5.38. 

3. Find the ^227 to A^nthin less than YTSm' ^^^^' ^^•^^^• 

Remark. — The number of ciphers to be annexed to the whole 
number, is always double the number of decimal places required 
to be found in the root. 

98. The manner of extracting the square root of a number 
containing an entire part and decimals, is deduced immediately 
from the preceding article. 

Let us take for example the number 3.425. This 5 equiva- 

3425 
lent to ^ . Now, 1000 is not a perfect squares, but the de- 



128 ELEMENTS OF ALGEiSRA. [CHAP, vi 

nominator may be made such without altering the value of the 
fraction, by multiplying both terms by 10 ; tliis gives 

34250 34250 

or 

10000 (100)2 

Then, extracting the square root of 34250 to the nearest unit, 

we find 185 ; hence, -— — or 1.85 is the required root to with- 
in less than — — . 

If greater exactness be required, it will be necessary to annex 
to the number 3.425 as many ciphers as shall make the num- 
ber of periods of decimals equal to the number of decimal 
places to be found in the root. Hence, to extract the square 
root of a mxixed decimal : 

Annex ciphers to the proposed number until the whole nmnler 
of decimal places shall be equal to double the number required in 
the root. Then^ extract the root to the nearest unit, and point off, 
from the right hand, the required number of decimal places. 

EXAMPLES. 

1. Find the y/ 3271.4707 to within less than .01. 

Ans. 57.19. 



2. Find the ^ 31.027 to within less than .01. Ans. 5.57. 

3. Find the ^ 0.01001 to within less than .00001. 

Ans. 0.10004. 

99 1 Finally, if it be required to find the square root of a 
vulgar fraction in terms of decimals : 

Change the vulgar fraction into a decimal and continue the di- 
vision until the number of decimal places is double the number 
required in the root. Then, extract the root of the decimal by th$ 
last rule. 



EXAMPLES. 



1. Extract the square root of — - to within less than .001 

14 

This number, reduced to decimals, is 0.785714 to within less 
than 0.000001. The root of 0.785714, to the nearest unit, is 



CHAP, v.] SQUARE ROOT OF ALGEBRAIC QUANTITIES. 129 

886 : hence, 0.886 is the root of -— to within less than .001. 

14 

2. Find the ,/2?l to within less than 0.0001. Ans. 1.6931. 

Extraction of the Square Root of Algebraic Quantities, 

100. Let us first consider the case of a monomial. 

In order to discover the process for extracting the square 
root, let us see how the square of a monomial is formed. 

By the rule for the multiplication of monomials (Art. 42), 
we have 

(5a263c)2 = 5a^^c X ^a^^c = 2^a^h^c^ ; 

that is, in order to square a monomial, it is necessary to 
square its co-efficient^ and double the exponent of eacU letter. 
Hence, to find the square root of a monomial, 

Extract the square root of the co-efficient for a new co-efficient^ 
and write after this, each letter, with an exponent equal to its 
original exponent divided by two. 

Tlius, y/64a66* = ^aW ; for, 8^352 x 8^352 - Q^a%\ 

and, y 025a2Z/8c6 = 25a6*c3 ; for, {2bab*c'^f = 625a^^c^ 

From the preceding rule, it follows, that, when a monomial' 
is a perfect square, its numerical co-efficient is a perfect square, 
and every exponent an even number. 

Thus, 2ba'^b^ is a perfect square, but 98a6* is not a perfect 
•quare ; for, 98 is not a perfect square, and . a is affected witli 
an uneven exponenL 



Of Polynomials. y^ 



101 • Let us next consider the case of polynomials. 

Let iV denote any polynomial whatever, arranged with refer- 
ence to a certain letter. Now the square of a polynomial is 
the product arising from multiplying the polynomial by itself 
©nee : hence, the frst term of the product, arranged iwith refer- 
ence to a particular letter, is the square of the first term of 
the polynomial, arranged witn reference to the same letter. 



130 ELEMENTS OF ALGEBRA. [CHAP. V. 

Therefore, the square root of the first term of such a product 
will be the first term of the required root. 

Denote this term by r, and the follo^^ing terms of the root, 
arranged with reference to the leading letter of the polynomial, 
by r\ r'\ r"\ &;c., and we shall have 

N^ (r + r' + r" + r'" + &c.;)2 
or, if we designate the sum of all the terms of the root, after 
the first, by 5, 

N —{r ■\-sY = r'^-\- 2rs + s^ 

= 7-2 + 2r (r' + r" + r'" + &c.) + s^. 
If now we subtract r^ from N, and designate the remaindei 
by -fi, we shall have, 

]Sr-r^ = B = 2r{r'-\- r" + r"' + &c.) + s2, 
which remainder will evidently be arranged with reference to 
the leading letter of the given polynomial. If the indicated 
operations be performed, the first term 2rr' will contain a 
higher power of the leading letter than either of the following 
terms, and cannot be reduced with any of them. Hence, 

If the first term of the first remainder be divided by twice the 
first term of the root^ the quotient will be the second term of 
the root. 

If now, we place r -\- r' z=: n, 

and designate the sum of the remaining terms of the root, 
r", r"\ &c., by s\ we shall have 

iV^= (?i + s')2 =zn^ -\- 2ns' + s'2. 
If now we sul)tract n^ from JV, and denote the remaindcT 
by R', we shall have, 

jSr-n^ = E' = 2ns' + s'2 ^ 2(r + r') (r" + r'" + &c.) -f s'>; 
in which, if we perform the multiplications indicated in the 
second member, the term 2rr" Mill contain a liigher power of 
the leading letter than either of the following terms, and can- 
not, consequently, be reduced with any of them. Hence, 

If the first term of the second remainder be divided by twice 
the first term of the root, the quotient will be the third term 
of the root. 



CHAP, v.] SQUARE ROOT OF ALGEBRAIC QUANTITIES. Ul 

4- r'^ + &c. = s". 



11 


we 


make 
















r + r' 


+ r" = 


< 


and 


r'" 


we 


shall 


have 
















li-- 


= {n' 


+5 


y 


= w'2 


+ 2; 



JSr-n'^ = R" == 2 (r + r' + r") {r'" + r" + &c.) -f «"2; 
in which, if we perform the operations indicated, the ternai 
irr'" will contain a higher power of the leading letter than 
any following term. Hence, 

If we divide the first term of the third remainder hy twice 
the first term of the root, the quotient will be the fourth term 
of the root. 

If we continue the operation, we shall see, generally, that 

The first term of any remainder^ divided hy twice the first 
term of the root, will give a new term of the required root. 

It should be observed, that instead of subtracting n^ from 
the given polynomial, in order to find the second remainder, 
that that remainder could be found by subtracting (2r + r')r' 
from the first remainder. So, the third remainder may be found 
by subtracting {2n + r")r" from the second, and similarly for 
the remainders which follow. 

Hence, for the extraction of the square root of a polynomial, 
we have the following 

RULE. 

I. Arrange the polynomial ivith reference to one of its letters, 
ind then extract the square root of the first term, which will give 
the first term of the root. Subtract the square of this term from 
the given polynomial. 

II. Divide the first term of the remainder hy twice the first term 
of the root, and the quotient will be the second term of the root. 

\f\. From the fi.rst remainder subtract the product of twice the 
iiriP^ term of the root plus the second term, by the second term. 

IV. Divide the first term of the second remainder by twice the 
hrst term of the root, and the quotient will be the third term q/ 
Uh' root. 



132 ELEMENTS OF ALGEBRA. LCHAP. V, 

V. From the second remainder subtract the product of tioice the 
sum. of the first and second terms of the root, plus the thv.'d 
term, by the third term, and the result will be the third remain- 
der, from which the fourth term of the root may be found as 
hefore. 

VI. Continue the operation till a remainder is found equal to 
0, or till the first term of some remainder is not divisible by 
twice the first term of the root. In the former case the root found 
is exact, and the polynomial is a perfect square; in the latter 
case, it is an imperfect square. 

EXAMPLES. 

1. Extract the square root of the polynoniial 

49a262 _ 24a63 ^25 a*- SOa^ + 166*. 

First arrange it with reference to the letter a. 



25a* - 30a36 + 49aW - 24aP + 166* 
25a* 



n = - 30a36 + 49a262 - 24a63 + 166^ 
-30a36+ 9a262 



R' = + 40a262 - 24a63 + 166* 

+ 40a262 - 24a63 + 166* 



R" = 



5a2 - 3a6 + 46^ 



10a2- 


-3a6 
-3a6 


- 30a36 + 9a26'-^ r 


10a2 - 


- 6a6 4- 462 
462 



40a262 - 24a63 + 166*. 



2. Find the square root of 

a* + 4:a^x + 6a^x^ + 4:ax^ 4- x\ 

3. Find the square root of 

a* — 2a^x -f 3a2a;2 — 2ax^ + x\ 

4. Find the square root of 

4a:6 4- 12:^5 + 5a:* - 2x3 ^ 7^2 _ 2x + 1. 

5. Find the square root of 

9:i* - 12a36 + 2Sa^^ - IQab^ + 166*. 

0. Find the square root of 
»^5a*62 — 40a362c -f 76a262c2 — 48a62c3 + 3662c* ~ 30^6c -4- 24a'6|pV 
\ \ - 36a26c3 + 9a*c2. ^^'^ % - ,5 



./ 



CHAP, v.] RADICALS OF THE SECOND DEGREE. 133 

Remarks on the Extraction of the Square Root of Pohjnomials. 

1st. A binomial can never be a perfect square. For, its root 
cannot be a monomial, since the square of a monomial will 
be a monomial ; nor can its root be a polynomial, since the 
square of the simplest polynomial, viz., a binomial, \dll cou 
tain at least three terms. Thus, an expression of the form 

a2 ± 62 
can never be a perfect square. 

2d. A trinomial, however, may be a perfect square. If so, 
when arranged, its two extreme terms must be squares, and the 
middle term double the product of the square roots of the other 
two. Therefore, to obtain the square root of a trinomial, when 
it is a perfect square, 

Extract the square roots of the two extreme terms, and give these 
roots the same or contrary signs, according as the middle term is 
vositive or negative. To verify it, see if the double product of the 
two roots is equal to the middle term of the trinomial. 

Thus, 9a^ — 48a*62 _^ 64rt26* is a perfect square, 
for, y'^Q^ = 3a3 ; and, ^ QAa^^ = — Sab^ ; 
also, 2 X 3a3 x(— Sab^-) = — ^Sa'^b^ the middle term. 

But 4a2 + Uab + 9b^ 

is not a perfect square : for, although 4^^ and -f- 9b^ are per- 
fect . squares, having for roots 2a and 35, yet 2 X 2a x 36 is 
not equal to lAab. 

Of Radical Quantities of the Second Degree. 

102. A radical quantity is the indicated root of an imperfect 
power of the degree indicated. Radical quantities are some- 
times called irrational quantities, sometimes surds^ but more 
commonly, simply radicals. 

The indicated root of a perfect power of the degree indi 
cated, is a rational quantity expressed under a radical form. 



134 ELEMENTS OF ALGEBRA. [CHAP. 



1 



An indicted square root of an imperfect square, is called 
a radical of the second degree. 

An indicated cube root of an imperfect cube, is called a radi- 
cal of the third degree. 

Generally, an indicated n^'^ root of an imperfect n^^ power, 
is called a radical of the n^^ degree. 

Thus, .V2J fW and y^, are radicals of the second degree ; 
^4, ^/Ts" and ^/TT, are radicals of the third degree; 
and \/4^ \/~^ and 'i/TT, are radicals of the n^^ degree. 
The degree of a radical is denoted by the index of the 
root. 

The index of the root is also called the index of the radical. 

103« Since like signs in both factors give a plus sign in tho 
product, the square of — «, as well as that of + a, will be 
tt^ : hence, the square root of a^ is either -\- a or — a. Also, 
the square root of '2ha%^ is either + 5«^^ or — hab"^. Whence 
we may conclude, that if a monomial is positive, its square root 
may be affected either with the sign -f or — ; 

thus, -y/^ = ± 3a2, 

for, + Sa^ or — Sa^, squared, gives 9a*. The double sign ± 
with which the root is affected, is read plus or minus. 

If the proposed monomial were negative^ it would have n(p 
square root, since it has just been shown that the square of every 
quantity, whether positive or negative, is essentially positive. 

Therefore, such expressions as, 

are algebraic symbols which indicate operations that cannot be 
performed. They are called imagina-^y quantities^ or rather, 
imaginary expressions^ and are frequently met mth in the so- 
lution of equations of the second degree. Generally, 

Every indicated even root of a negative quantity is an imaginary 
expression. , 

An odd root of a negative quantity may oflen be extracted 
Fo- example, y — 27 = — 3, since (—3)3=— 27. 



CHAP, v.] KAPICALS ;F THE SECOND DEGREE. 136 

Radicals are similar when they are of the same degree aud 
the quantity under the radical sign is the same in both. 

Thus, CL yfh and c ,J~b^ are similar radicals of the second 
degree. 



Of the Simplification of Radicals of the Second Degree. j 

104. Radicals of the second degree may often be simplified, 
and otherwise transformed, by the aid of the following prin- 
ciples. 

1st. Let the y/^ and ,V^ denote any two radicals of the 
second degree, and denote their product by p\ whence, 
^X^b=p .... (1). 

Squaring both members of equation (1), (axiom 5), we have, 

or, ab^p^ - - - - (2). 

Extracting the square root of both members of equation (2), 
(axiom 6), we have, 

y/ab=p', 
but things which are equal to the same thing are equal to eacli 
other, whence, 

y/~a X^ = .^J~ab; hence. 
The product of the square roots of two quantities is equal to 
the square root of the product of those quantities. 

2d. Denote the quotient of .Va by V^^ by q ; whence, 

jt" ■■■■ <■> 

Squaring both members of equation (1), we find, 

or, ^=:j2 . . . . (2). 

Extracting the square root of both members of equation (2), 
we have, 

\ — ^^ 



136 ELEMENTS OF ALGEBRA. [CHAP. V. 

Things which are equal to the same thing a-e equal to each 
other, whence, 



y -^^', hence, 



■ The quotient of the square roots of two quantities is equal to 
tJie square root of the quotient of the same quantities. 

105i The square root of 98a6* may be placed under the form 
y/98^=y49/;-^ X 2a, 
which, from the 1st principle above, may be written, 
^496* X y^2^= 7^*2^2^, 
In like manner, 

^ A:haWcM =^^aWc^ X hhd — Zabc^fM. 
^864a265cii r=y 144a26^cio X ^hc = 12a?/Vy66c. 
The quantity which stands without the radical sign is called 
tiie^ co-efficient of the radical. 

Thus, 752, Sabc, and 12ab'^c^, are co-efficients of the radicals. 
In general, to simplify a radical of the second degree : 
I. Resolve the quantity under the radical sign into two factors^ one, 
of which shall be the greatest perfect square which enters it as a factor. 

II. Write the square root of the perfect square before the radical 
sign; under which place the other factor. 



EXAMPLES. 



'\^'> M\ 



1. Reduce ^lba%c to its simplest form. 



2. Reduce J 12Sb^a^d^ to its simplest form. 

0. Reduce ^ S'2a^b^c to its simplest form. 

4. Redace V 2b^a'^b'^c^ to its simplest form. 

5. Reduce yi02-^a^'c^ to its simplest form. 



6. Reduce V T2Sa'^b^c^d to its simplest form. 

If the quantity under the radical sign is a polynomial, we 
may often simplify the expression by the same rule. 



CHAP, v.] RADICALS OF THE SECOND DEGREE. 137 

Take, for example, the expression 



Tlie quantity under the radical sign is not a perfect square : 
but it can be put under the form 

ah [o? + 4.ab + 4^2). 
Now, the factor within the parenthesis is eadently the square 
of a + 26, whence we have 

y^6T4a26M^i^6^ = (a + 26) y^^ 
105*t Conversely, we may introduce a factor under the radical 
sign. 

Thus, «V^=V^"-/^ 

which by article 104, is equal to J~^ Hence, 

The co-efficient of a radical may he passed under the radical sign, 
as a factor^ by squaring it. 

The principal use of this transformation, is to find an ap- 
proximate value of any radical, which shall differ from its true 
value, by less than 1. 

For example, take the expression 6^13. 
" Now, as 13 is not a perfect square, we can only find an ap- 
pi'oximate value for its square root ; and when this approximate 
value is multiplied by 6, the product will differ miaterially from 
the true value of 6^13. But if we write, 

6yi3 ^y02 X 13 =^36 X 13 =:y/468, 
we find that the square root of 468 is the whole number 21, 
to within less than 1. Hence, 

6,yA3 = 21, to within less than 1. 
In a similar manner we may find, 

12 /T =31, to within less than 1. 

Addition and Subtraction. ^ 

106t In order to add or substract similar radicals : 
Add or subtract their co-efficients^ and to the sum or differ^ 
ence annex the common radical. 



138 ELEMENTS OF ALGEBRA. LCHAP. V 



Thus, Za^ +bc^h =(3a-f5c)yT; 

and Za^ - bc^T = (3a - 5c) ^6^ 

In like manner, 

7^a + 3 ^2^ = ("7 + 3)y^ = lO^a ; 
and . 7y^ - 3 ^2^ = [^ - 3)y^ = 4^/2^. 

Two radicals, which do not appear to be similar, may becumi 
80 by simplification (Art. 104). 
For example, 

Also, 2 y45 - 3yy = 6 ^5" - 3 ^5" = 3 ^5. 

When the radicals are not similar, their addition or subtrac- 
tion can only be indicated. 

Tims, to add 3.yA6 to SV^, we write, 

5^7+ 3-/6. 

Multijplication of Radical Quantities of the Second Degree, 

107. Let a.^rb and cJd denote any two radicals of the second 
degree; their product will be denoted thus, 

which, since the order of the factors may be changed without 
altering the value of the product, may be wi'itten, 

axe Xy^Xy^ 
The product o' the last factors from the 1st principle of Art 
104, is equal .o ,JTd-, we have, therefore, 

dyfb X cJTi — acJhd. 

Hence, t" multiply one radical of the second degree by au 
cither, we have the following 

RULE. 

Multiply the co-efficients together for a new co-efficient; after this 
write the radkal sign, and under it the product of the juantitiet 
under both radical signs. 



i 



CHAP, v.] RAEICALS OF THE SECOND DEGREE. 139 



EXAMPLES. 



1. 3^506 X 4^'^20a = 12yi00a2? = 120ayT. 



3. 2a^ a' + b^ X - Sa^a^ + 62 = - Ga^ (a^ + b^). 

Division of Radical Quantities of the Second Degree. 

108* Let (i.Jh and c.J~d represent any two radicals of the 
second degree, and let it be required to find the quotient of the 
first by the second. This quotient may be indicated thus, 

--^f-—. which IS equal to — X ^ . ; 
c/rf ^ 

but from the 2d principle of Art. 104, 

y^ _ /T «y^ a fb 

r—- ~\/ T\ hence — p: = — \/ —• 

^ V d ' c^d c V d 

Hence, to divide one radical of the second degree by another, 
we have the foil owning 

RULE. 

Divide the co-e^cient of the dividend hy the co-efficient of the 
divisor for a new co-efficient ; after this, write the radical sign^ 
placing under it the quotient obtained by dividing the quantity under 
the radical sign in the dividend by that in the divisor. 

For example, ^ajb • ^bJ~c^^ lT-k\ — » 
* ^ ' lib V c 

And, \2ac^/&^-^ Ac^b = Sa \/-^ = 3a y^." 

109. The following transformation is of frequent application ia 
finding an approximate value for a radical expression of a par- 
ticular form. 

Having giren an expression of the form, 

a a 
— or — , 



140 ELEMENTS OF ALGEBRA. [CHAP. V 

in which » and p are any numbers whatever, and q not a per 
feet square, it is the object of the transformation to render the 
denominator a rational quantity. 

Tliis object is attained by multiplying both terms of the frac 
tion by i? — -/^ when the denominator is p-i-y/~q, and by 
p -\- .,y~q, when the denominator is p — ^9', and recollecting 
that the sum of two quantities, multiplied by their difference, i» 
equal to the difference of their squares : hence, 

a _ o-{p + -y/g) _ ojp -\- V~q) _ g/> + a^/~q 

in which the denominators are rational. 

As an example to illustrate the utility of this method of ap- 
proximation, let it be required to find the approximate value of 

the expression ^=. We write 

7 _ 7(3 + lAy _ 21 + 7 -/5 

3 - y y ~ 9 - 5 ~ 4 

But, "i ^^ ^49 X 5 = ^245 == 15 to within less 

than 1. Therefore, 

7 21 + 15 to within less than 1 

— — = 9 to withm 



3-^5 4 

less than — • ; hence, 9 differs from the true value by less than 
one fourth. 

If we wish a more exact value for this expression, extract ike 
square root of 245 to a certain number of decimal places^ add 21 
to this root^ ana divide the result by 4. 

Take the expression, . — ,-^^- 

and find its value to witliin less than 0.01. 



OHAP. v.] EXAMPLES IN THE CALCULUS 01 KAJDICALS. 141 
We have, 

7/ 5 7/5 (-/n - '/ 3)"_ 7/55 - 7yi5 

/Tl + ys" 11-3 "" 8 

Now, 7^55 =^55 X 49 =^2695 = 51.91, within less than 0.01, 
and 7y/i5=yi5x 49=^/735 =27.11; - J -y f / 
therefore, — r--- 

7-/5~ 51.91-27.11 24.80 ^^. ''^ ^*" 

= 0.10. 



ll+ys 8 8 

Hence, we have 3.10 for the required result. This is true to 



within less than — — . 
bOO 

By a similar process, it may be found, ihat, 

3 _)_2n/7' 

—r: — - — — =2.123, is exact to within less than 0;'O01. 

5/12-6/5 

Remark. — The value of expressions similar to those above, 
may be calculated by approximating to the value of each of the 
radicals which enter the numerator and denominator. But as 
the value of the denominator would not be exact, we could 
not determine the degree of approximation which would be 
obtained, whereas by the method just indicated, the denomina- 
tor becomes rational, and we always know to what degree of 
accuracy the approximation is made. 

PROMISCUOUS EXAMPLES. 

1. Simplify / 125. Ans. 5/5. 

2. Reduce x/rrr; to its simplest form. 



We observe that 25 will divide the numerator, and hence, 

2^ 

147 

Since the perfect square 49 will divide 147, 



^50" _ /\ 

147 ~ V ~ 



V 147 ~ V 49 X 3 ~ 7 V 3 



142 



ELEMEJsTS OF ALGEBEA. 



[CHAP. V. 



Divide the coefficient of tlie radical by 3, and mu tiply tlie num 
ber under the radical by the square of 3 ; then, 



i\j~ 21 Vt ~ 21'^ ' 



3. Reduce ^ QSa'^x to its most simple form. 

Ans. 7aV2ir. 

4. Reduce ^ (x^ — a^x^) to its most simple form, /t l^'^CxA ^ 

5. Required the sum of ^72 and Vl28. 

Ans. 14/2? 

6. Required the sum of .J~^ and / 147. 

Ans. 10^/3: 



/2" 



7. Required the sum of \ / — 



and 



A... f^^. 



8. Required the sum of 2 V a?h and 3 V 646.2;*. 

9. Required the sum of 9^243 and 10^363. 

3" , /T 

27* 



Yt 



"A I 

10. Reauired the difference of \/-=- and \/ 

V 5 V ; 



^?1S. 



45 



11. Required the product of SVIT and 3^5. 

Ans. 30^/Ta 



'"""^"^ 2 /HT 3 

12. Required the product -of -7r\/-^ and — . . ,^. 

3 V 8 4 V 10 



Ans. 



To^'- 



13. Divide eyiO by 3^5] ^ Ko^ 

14. What is the sum of ^ 48a62 and h.^J^a. ^ A^ 

15. What is the suir of ^TS^^ and y^SOa^R 

^l ^??5. (3a26 + 5a6)^ 2aX 

W 



/6^ 4i 



910/-- 



CHAPTER n 

EQUATIONS OF THE SECOND I COUCC. 

llOt Equations of the second degree may involve bui on$ 
unknown quantity, or they may involve moj-e than one. 
We shall first consider the former class. 

Ill, An equation containing but one unknown quantity is 
said to be of the second degree, when the highest power of 
the unknown quantity in any term, is the second. 

Let us assume the equation, 

-z-x^ — ex -^ d = cx"^ -\ — -X -{- a, 
b d 

Clearing of fractions, 

adx"^ — bcdx + bd^ = bcdx"^ -f- b^x -\- abd 

transposing, adx"^ — bcdx^ — bcdx — b^x = abd — bd"^ 

factoring, {ad — bcd)x'^ — {bed -f b'^)x = abd — bd"^ 

dividing both members by the co-efficient of x^^ 

bcd-{-b^ ahd-hd^ 

x^ X =. . 

ad — bed ad — bed 

If we now replace the co-efficient of x by 2/>, and th« 
second member by q^ we shall have 

x^ -f 2px = q ', ■ 
and since every equation of the second degree may be reduced, 
iu like manner, we conclude that, every equation of the second 
degree, involving but one unknown quant'^y, can be reduced to 
the form 

x^ -h ^px = g, 
by the following 



144 ELEMENTS OF ALGEBRA. [CHAP. VI. 

RULE. 



L Clear the equation of fractions ; 

n. Transpose all the known terms to the second member^ and 
all the unknown terms to the first. 

ni. Reduce the terms involving the square of the unknown 
quantity to a single term of two factors^ one of which is the 
square of the unknown quantity ; 

IV. Then^ divide both members by the co-efficient of the square 
of the unknown quantity. 

112. If 2/?, the algebraic sum of the co-efficients of the first 

powers of a;, becomes equal to 0, the equation will take the 

form 

x' = q, 

and this is called, an incomplete equation of the second degree. 

Hence, 

An i7icomplete equation of the second degree involves only the 

second power of the unknown quantity and known terms, and map 

be reduced to the form 

X^ =: q. 

Solution of Incomplete Equations. 

113. Having reduced the equation to the required form, we 
have simply to extract the square root of both members to find ilie 
value of the unknown quantity. 

Extracting the square root of both members of the equation 

x"^ =z q, we have x = -/^ 

If 5- is a perfect square, the exact value of x can be found 
by extracting the square root of q, and the value of x will then 
be expressed either algebraically or in numbers. 

If q is an algebraic quantity, and not a perfect square, it must 
be reduced to its simplest form by the rules for reducing radir 
cals of the second degree. If g' is a number, and not a perfect 
square, its square root must be determined, approximately, by 
the rules already given. 



4 



CHAP. VI.l EQUATIONS OF THE SECOND DEGREE. 145 

But the square of any number is -f-, whether the number 
itself have the -f- or — sign; hence, it follows that 

(+V^? = ?. and (-y^)2-:r, 
and therefore, the unknovrn quantity x is susceptible of two dis- 
tinct values, viz : 

« = +^/^ and z = -yT; 
and either of these values, being substituted for x, will satisfy 
tlie given equation. For, 

and a;2 = —J~q X —J~q-=z q\ hence, 

Every incomplete equation of lite aecoud degree has two rooti 
which are numerically/ equal to each other; one having the sign^ 
plus, and the other the sign minus (Art. 77). 



EXAMPLES. 

1. Let us take the equation 

which, by making the terms entire, becomes 

Sx^ - 72 + 102;2 = 7 - 24^-2 + 299, 
and by transposing and reducing 

42a:2 = 378 and x^ = ^ = 9 ; 

hence, x =. -|--/9"= + 3 ; and x = — .^^= — 3. 

2. As a second example, let us take the equation 

3a;2 = 5. 
Dividing both members by 3 and extracting the square root, 

X z=z ± \ — = rh — ^/ 15: 

V 3 ^y ' 

In which the values of x must be determined approximately 

3. What are the values of x in the equation 

11(.^2 _ 4) rrz 5(a:2 _|. 2). Ans, ar = ± 3.. 

4. What are the values of x in the equation 

'J m^ — x^ . m 

■ = n. Ans X = ±: 



X 



y 1 4- n^ 



146 ELEMENTS OF ALGEBRA. [CHAP. VI. 

Solution of Equations of tJie Second Degree, 

114. Let us now solve the equation of the second degree 

x^ + ^px z=iq. 
If we compare the first member with the square of 
X -\- p^ which is x"^ -\-2px -\- p^, 
"we see, that it needs but the square of p to render it a perfect 
square. If then, jp^ be added to the first member, it will be 
come a perfect square ; but in order to preserve the equality of 
the members, p^ must also be added to the second member. 
Making these additions, we have 

x"^ + 2px -\- p"^ z=z q -\- p^ '^ 
this is called, completing the square, and is done, by adding tnje 
square of half the co-efficient of x to both members of the equa 
Hon. 

Now, if we extract the square root of both members, we have, 

x+p= ±:^q -{-p% 
and by transposing p, we shall have 



X = —p -\-^ q +i?^ and x = —p —^q+p"^. 
Either of these values, being substituted for x in the equation 

x"^ -\- 2px = q 
will satisfy it. For, substituting the first value, ^ 

x^ = {-p -{-/iTp^y =p^ -2p./'^Tp^ -{- q -^ p\ 
and 



2px = 2px{ -p-\-^q+p^) = - 2i?2 -j- 2p^q-\-p^, 
by adding x^ + 2px = q. 

Substituting the second value of a;, we find, 

x^ = {-p -^7+^)2 ^ ^24. 2p/7T^4- q -f r^ 
and 

2px ^2p{-p -^7+72) =-2p^- 2^y^7Tp ; 
by adding x^ + 2px = q ; 

and consequently, both values found above, are roots of the 
equation. 



CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 147 

In order to refer readily, to either of these values, we shall 
call the one which arises from using the + sign before the 
radical, the first value of a;, or the first root of the equation; 
and the other, the second value of x, or the second root of the 
equation. 

Having reduced a complete equation of the second degree to 
the form 

a;2 + 2px = q, 
we can write immediately the two values of the unknown quan 
tity by the following 

RULE. 

I, The first value of the unknown quantity is equal to half 
the co-efficient of «, taken with a contrary sign, plus the square 
root of the second memher increased by the square of half this 
co-efficient. 

II. The second value is equal to half the co-efficient of a;, 
taken with a contrary sign, minus the square root of the second 
member increased by the square of half this co-efficient, 

EXAMPLES. 

1. Let us take as an example, 

a;2 _ 7a: + 10 = 0. 
Reducing to required form, 

a:2-7a;=: -10; 

whence by the rule, x = -— -{-</— 10 -\ = 5; 

7 / Zq 

and, x = --y'-10 + ^ = 2. 

2. As a second example, let us take the equation 

6 2'4-° 3 +l2* 



1 is ELEMENTS OF ALGEBRA. [CHAP. VL 

Reducing tc the required form, we have, 

2 _ 360 
"^ '^22''-"22'' 



whence, .= -l+/57(J)' 



It often occurs, in the solution of equations, that p^ and q 
are fractions, as in the above example. These fractions most 
generally arise from dividing by the co-efficient of x^ in the 
reduction of the equation to the required form. When this is 
the case, we readily discover the quantity by which it is neces- 
sary to multiply the term q, in order to reduce it to the 
same denominator with p^ ; after which, the numerators may be 
added together and placed over the common denominator. 
After this operation, the denominator will be a perfect square, 
and may be brought from under the radical sign, and will 
become a divisor of the square root of the numerator. 

To apply these principles in reducing the radical part of the 
values of x, in the last example, we have 



/360 / 1 y_ . / 360 X 22 1 _ /7920 + 1 

V 22 "^ V22/ ~ V (22)2 + (22)"2 " " V (22)^ 

and therefore, the two values of x become, 

'^~~22'^22~22~ ' 

_ 1 89 90 45 

and 0:=---- = --=--; 

either of which being substituted for x in the given equation, 
will satisfy it. 

3. What ar« the valufcs of a; in the equation 
az^ — ac z= ex — bz^ 



CHA.P. VI.] EQUATIONS OF THE SECOND DEGREE. 149 

Reducing to required form, we have, 

c ac 



a -\- b a -f- 6 



C / Q.C C' 

whence, x =z -\- —-, — + \ / — — ^ + 



2{a^b) ' V a + 6 ' 4 (a + 6)2' 
= + 2(a + 6)~Vc 



and. 



(a + 6) V a + 6 4 (a + 6)2' 
Reducing the terms under the radical sign to a common 

denominator, we find, 

/ ac ~~ c2 /4a2c -\-^i-nhr -\- c2 -Y/4a2c -f- 4a6c + e^ 

V ^+1 ^ 4 (a + 6)2 "^ V ~4 (a +1)2 ^ 2 (a + 6) ' 

, cdb V 4a2c -f 4a6c + c2 

hence, x = ^—7^^ tt . 

2{a + b) 

4. What are the values of x, in the equation, 

62^2 — 37ar = — 57. 
By reducing to the required form, we have, 
. 37 57 



, 37 / 57 , /37\2 

wnence, x = -\ ±\/ hi — I 

^ 12 V 6 ^\12/ 

Reducing the quantities under the radical sign to a common 
denominator, we have, 



37 /^TirxlfTTW 

^12 V (12)2 ■^(12)2- 

But, 114 X 12 = 1368 ; and (37)2 _ 1359 . 

, , 37 , /- 13G8 -f- 1369 37 . 1 

hence, ^=+t7^±\/ ^^^ = A rt — 

' ^ 12 V (12)2 ^ 12 12 



<>'» « = + tt: +• T?: = — , 



(12)2 '12 12' 

37 1_19 
12 "^ 12 ~ 6 



A . 37 1 ^ 

^'' ^=+12-12 = ^- 

5. What are the values of x, in the equation, 
4a2 - 2x°- + 2ax = 18a6 - 1S6=». 



150 ELEMENTS OF ALGEBRA. [CHAP. VX 

Reducing to the required form, we have, 
«2 _ aa; = 2a^ — 9ab + 9b^ ; 



whence, » = — db \/ 



2a2 - 9ab + 9b^ +-- 
4 



|±^^^-9a5 + 95. 



The radical part is equal to — Sb; hence, 



36) ; or i 
Find the values of a; in the following 



a ,2a 
2 ^2 



EXAMPLES. 



x^ a ^ b 2x'^ ^ a b 

I. ~ — -j-x=l X -. Ans. x= ~, xz=z . 

6 a 6 a 



. (/a: 3a;2 1 + c x"^ ^ x 



a;2 2a; m x"^ x 

^' 4 -y^-S-^^-T "3 



^?i5. a: = -r, a? = 



3 S 

^n.. a;=-, ar = - - 



t 



^ 90 90 27 , ^ ^ 5 

4. — - = — — -. Ans. a: -- 4, x = — —- 

a;a;+la;-h2 3 



5. ?^_2 = =±1. Ans. .=7, .= 1 

8 — a; X — '2 5 

a;2 , , 6 - 1 , , . 6 , 6 

6. ax --\- b =. — ; — x^ + — rr. Ans. x — a. x = " 

boa a 

^ a — 6 . 8^2 ap. j _}_ ^ ^2 52 

e 2 c^ c 2 c^ 

. b -\- a b — a 

Ans. X = , X = , 

c c 



4 






? 3 L ^?l^ 



CBULP. VI.] EQUATIDJJs OF THK' SECOND DEGREE ' ^ ;' 151 



Ans. X = 



mn 



X = 



.yf^—yf^ y^+yn 



9 aiar? ~ -\- = — ^ x. 

c^ c c^ c 

2a —b 

Ans. X = , X = 

ac 



,^ 4a;2 , 2a; , ,. ,^ 3.^2 58;^; 

iO. -;;^ +-X-+10 = 19-— -4- -^. 

7 7 7 7 



3a -\- 2b 
be • 



X 4- a a — X 

X — a 



Ans, X = 9, X = — 1. 
Ans. X 



±ayj- 



b + 2 



6-2 



a -{- X 

12. 2a;4-2 = 24 — 5a: — 2.^2, Ans. x = 2, x 

13. x^ — X'-40 = 170. ^715. X = 15, and a; =: — 14. 



11 
2' 



y 14. 3a;2 + 22: — 9 = 76. Ans. a; = 5, and a; = — 5f . 

15. a2 + 62 _ 26a: + a;2 z= — — . 



Ans. X = —= (bn ± ^/ci^m^ -\- b'^m'^ — a^n'^), 

n^ — nfi ^ V ' 




Problems giving ris( to Equations of the Second Degree invohf* 
ing hut one unknown quantity. \ 

1. Find a number such that three times the number added to 
twice its square will be equal to G5. 

Let X denote the number. Then from the conditions, 

2a;2 + 3a;=65 - - - (1) 

Whence, a: = -J±y/^ + ^; 

reduciog 



13 



X =>. b and r = — — - 



152 ELEMENTS OF ALGEBRA. LCHAP IV. 

Both of these roots verify the equation : for, 

2 X (5)2 + 3 X 5 = 2 X 25 + 15 = 05; 

J o/ 13\2 , ^ 13 169 39 130 ^^ 

and 2(--)+3x--^^ — --= — = 6o. 

The first root satisfies the conditions of the problem as enui» 
eiated. 

The second root will also satisfy the conditions, if we regard 
its algebraic sign. Had we denoted the unknown quantity by 
— X, we should have found 

2x^-Sx = 65 - - - (2) 

13 

from which x z=z — and x = — 5. 

We see that the roots of this equation differ from those of 
equation (1) only in their signs, a result which was to have 
been expected, since we can change equation (1) into equation 
(2) by simply changing the sign of x^ and the reverse. 

2. A person purchased a number of yards of cloth for 240 
cents. If he had received three yards less, for the same sum, it 
would have cost him 4 cents more per yard. How many yards 
did he purchase 1 

Let X denote the number of yards purchased. 

240 
Then will denote the number of cents paid per yard. 

Had he received three yards less, 
a; — 3, would have denoted the number of yards purchased, and 

240 

-, would have denoted the number of cents he paid per j^aid* 

2^ — o 

From the conditions of the problem, 

240 240 _ 

a; - 3 ~ir- ' 

by reducing. x^ — 3:c = 180, 

whence, ar := 15 and a; = —12. 

The value .r = 15 satisfies the coniitions (f the problem, 
understood in their arithmetical sense; for, 1{ yards for 240 



CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 153 

240 
cents, gives -— — , or 16 cents for the price of one yard, and 

ID 

12 yards for 240 cents, gives 20 cents for the price of one 
yard, which exceeds 16 by 4. 

The value + a: = — 12, or -- a; = -f 12, will satisfy the 
conditions of the following problem : 

A person sold a number of yards of cloth for 240 '^.eiits : 
if he had received the same sum for 3 yards more^ it would 
have brought him 4 cents less per yard. How many yards did 
he sell? 

If we denote the number of yards sold by x^ the statement of this 
last problem, and the given one, both give rise to the same equation, 

x"^ —Zx = 180, 

hence, the solution of this equation ought to give the answers 
to both problems, as we see that it does. 

Generally, when the solution of the equation of a problem 
gives two roots, if the problem does not admit of two solu- 
tions there is always another problem whose statement gives 
rise to the same equation as the given one, and in this case 
the two roots form answers to both problems. 

3. A man bought a horse, which he sold for 24 dollars. At 
^he sale, he lost as much per cent, on the price of his pur- 
chase, as the horse cost him. What did he pay for the horse ? 

Let X denote the number of dollars that he paid for the horse : 

then, X — 24 will denote the number of dollars that he lost. 

But as he lost x per cent, by the sale, he must have lost 

X 

upon each dollar, and upon x dollars he lost a number 

x^ 
of dollars denoted by -—r ; we have then the equation 

— — = x — 24:, whence a;^ — lOO^r = — 2400 ; 

Therefore, x — QO and x — 40. 

Both of these values satisfy the conditions of the problem. 



154 ELEMENTS OF ALGEBRA. LCKAP. VI. 

For, in the first place, suppose the man gave 60 dollars for 
the horse and sold him for 24, he then loses 36 dollars. But, 
from the enunciation, he should lose 60 per cent, of 60, that is, 

loo "^^^ =-wr=^^' 

.berefore, 60 satisfies the problem. 

If he pays 40 dollars for the horse, he loses 16 by the sale ; 
for, he should lose 40 per cent of 40, or 

40 

therefore, 40 satisfies the conditions of the problem. 

4. A grazier bought as many sheep as cost him £60, and 
after reserving 15 out of the number, he sold the remainder 
for £54, and gained 25. a head^ j^ those he sold : how many 
did he buy ? «--^^ .^ Ans. 75. 

5. A merchant bought cloth for which he paid £33 155., which 
he sold again at £2 85. per/ piece, and gained by the bargain 
as much as one piece cost him : how many pieces did he buy ? 

Ans. 15. 

6. What number is that, which, bei^g divided by the product 
of its digits, the quotient will be 3 ; and if 18 be added to 
it, the order of its digits will be reversed 1 Ans. 24. 

7. Find a number such that if you subtract it from 10, apd 
multiply the remainder by the number itself, the product will 
be 21. Ans. 7 or 3. 

8. Two persons, A and B, departed from different places at 
the same time, and traveled towards each other. On meeting, 
it appeared that A had traveled 18 miles more than B ; and 
that A could have performed B's journey in 15| days, but B 
would have been 28 da)b in performing A's journey. How 
far did each travel 1 j A 72 miles. 

B 54 miles." 



■1 



9. A company at a tavern had £8 155. to pay for their 
reckoning ; but before the bill was settled, two of them left 



CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 155 

the room, and then those who remained nad lOs. apiece more 
to pay than before :' how many were there in the company 1 

Ans. 7. 

10. What two numbers are those whose difference is 15, and 
of which the cube of the lesser is equal to half their product ? 

Ans. 3 and 18. 

11. Two partners, A and B, gained $140 in trade: A's money 
was 3 months in trade, and his gain was $60 less than his 
"tock : B's money was $50 more than A% and was in trade 5 
months : what was A's stock "? Ans, $100. 

12. Two persons, A and B, start from two different points, and 
travel toward each other. When they meet, it appears that 
A has traveled 30 miles more than B. It also appears that 
it will take A 4 days to travel the road that B had come, 
and B 9 days to travel the road that A had come. What was 
their distance apart when they set outi Ans. 150 miles. 

Discussion of Equations of the Second Degree involving btii 
one unknown quantity. 

115. It has been sho\vn that every complete equation of the 
second degree can be reduced to the form (Art. 113) 

x^-{^2px = q - - - (1), 

in which p and q are numerical or algebraic, entire or frac- 
tional, and their signs plus or minus. 

If we make the first member a perfect square, by completing 
the square (Art. 112*), we have 

z^ + 2px -\- p^ =: q -\~ p^^ 
which may be put under the form 

Now, wha'^ver may be the value of q -\- p^, its square root 
may be represented by m, and the eouation put under the forio 

(ar -\- pY = m^, and consequently (x -f- pY — m^ = 0. 



156 ELEMENTS OF ALGLURA. [CHAP. VL 

But. as the first member of the last equation is the differenc* 
between two squares, it may be put under the form 

{x-{-p-m){x-\-p-{-m) = - - . (2), 
m which the first member is the product of t^o factors, and the 
second 0. No ,7, we can make this product eq\<al to 0, and 
con«!equently satisfy equation (2) only in two different ways . 
viz., by maldng 

X -{- p — m = 0, whence, x = — ^ -|- tti, 
or, by making 

X -{-p -^ m = 0, whence, x = — p — m. 
Now, either of these values being substituted for x in equa^ 
tion (2), will satisfy that equation, and consequently, will satisfy 
equation (1), from which it was derived. Hence, we conclude, 

1st. That every equation of the second degree has two roots, and 
only two. 

2d. That the first member of every equation of the second degree^ 
whose second member is 0, can be resolved into two binomial fac- 
tors of the first degree with respect to the unJcnown quantity, having 
the unknown quantity for a first term and the two roots, with their 
signs changed, for second terms. 

For example, the equation 

a;2 4- S^r — 28 = 
being solved, gives 

ic = 4 and x = —1 \ 
either of which values will satisfy the equation. We also have 
{^x - 4) (.r -f 7) = a;2 4- 3a: - 28 = 0. 

If the roots of an equation are known, we can readily form 
the binomial factors and deduce the equation. 

EXAMPLES. 

1. What are the factors, and what is the equation, of which 
the roots are 8 and — 9 ? 

Ans. ar — 8 and x ■\- ^ are the binomial factora, 
and a;2 + a; — 72 = is ;ne equat on. 



CHAF, VI.] EQUATIONS OF THE SECOND DEGREE. 157 

2. What are the factors, and what is the equation, of which 
the roots are — 1 and +11 

X -{- 1 and X — 1 are the factors, 

and a-2 — 1 = is the equation. 

3. What are the factors, and what is the equation, whose 
roots are 

7 + -v/ - i039 , 7 -V- 1039 „ 



are the factors, 

and Sx^ — 7x -\- S4: = is the equation. 

116i If we designate the two roots, found in the preceding 
article, by x' and x'\ we shall have, 

x' = — p -\- m. 



«" =z — jp — m; 



or substituting for m its value J q-\- p"^, 
x' = -p^^q-\-p2^ 

x" = —p—^q+pK 
Adding these equations, member to member, we get 
x' + x'' = —2p; 

and multiplying them, member by member, and reducing, 

we find 

x' x" = — g. 

Hence, after an equation has been reduced to the form of 

x^ + 2px = q^ 

1st. The ilgehraic sum of its two roots is equal to the co-efflr 
dent of the first power of the unhiown quantity^ with its sign 
changed. 

2d. The product of the two roots is equal to the second member^ 
unth its sign chan^'ed. 



158 ELEMENTS OF ALGEBRA. [CHAP. VI. 

If the sum of two quantities is given or known, their pro- 
duct will be the greatest possible when they are equal. I 

Let 2p be the sum of two quantities, and denote their diifei^ 
ence by 2d', then, 
p -\- d will denote the greater, and p — d the less quantity. 

If we represent their product by q, we shall have 
p"^ — d"^ — q. 

Now, it is plain that q will Increase as d diminishes, and 
that it will be the greatest possible, when c? = ; that is, when 
the two quantities are equal to each other, in which case the 
product becomes equal to p"^. Hence, 

3d. The greatest possible value of the product of the two roots, 
is equal to the square of half the co-efficient of the first power 
of the unknown quantity. 

Of the Four Forms. 

11 7i Thus far, we have regarded p and q as algebraic quan- 
tities, without considering the essential sign of either, nor have 
we at all regarded their relative values. 

If we first suppose p and q to be both essentially positive, 
then to become negative in succession, and after that, both to 
become negative together, we shall have all the combinations 
of signs which can arise. The complete equation of the second 
degree will, therefore, always be expressed under ore of the 
fo'jr following forms : — 

x2 -f. 2px = q (1), 
x2 -2px= q (2), 
x'^^pxz^ -q (3), 
a;2 - 2px = —q (4). 
These equations being solved, give 

x = -^±y~y+^ (1), 

^=-^P^y/ 9+P^ (2), 

« = -p^^Z-g+p"" (3), 

«= -fjt?±/^$"T7 (4). 



CHAP. VI.] EQUA'nONS OF THE SECOND DEGREE. 159 

In the first and second forms, the quantity under the radical 
sign will be positive, whatever be the relative values of ^ and g, 
since q and p^ ar« both positive ; and therefore, both roots 
will be real. And since 

q-^-p^yp"^, it follows that, ^q^p'^>P, 
and consequently, the roots in both these forms tvill have the same 
signs as the radicals. 

In the first form, the first root will be positive and the 
second negative, the negative root being numerically the greater. 

In the second form, the first root is positive and the second 
negative, the positive root being numerically the greater 

In the third and fourth forms, if , 

p^ > q, 

the roots will yc real, and since 

P>^/-(i-^P\ 
they will have the same sign as the entire part of the root 
Hence, both roots will be negative in the third form, and hot?, 
positive in the fourth. ^ 

If p"^ z=z q, the quantity under the radical sign becomes 0, 
and the two values of x in both the third and fourth forma 
will be equal to each other ; both equ^l to — p \n the third 
form, and both equal to -\- p in the fourth. 

If p"^ < q, the quantity under the radical sign is negative, 
and all the roots in the third and fourth forms are imaginary. 

But from the third principle demonstrated in Art. 110, the 
greatest value of the product of the two roots is p"^, and from 
the second principle in the same article, this product is equal 
to q ; hence, the supposition of p"^ <iq is absurd, and the values 
of the roots corresponding to the supposition ought to be inv 
possible or imaginary. 

When any particular supposition gives rise to imaginary re- 
suits, we interpret these results as indicating that the suppo 
sition is absurd or impossible. 



160 ELEMENTS OF ALGEBRA. [CHAP. VL 

If p = 0, the roots In each form become equal with con- 
trary signs ; real in the first and second forms, and imaginary 
in the third and fourth. 

If q = 0, the first and third forms become the same, as also, 
the second and fourth. 

In the former case, the first root is equal to 0, and tne 
second root is equal to — 2p ; m the latter case, the first root 
is equal to + 2p, and the second to , 0. 

If p =z and q = 0^ all the roots in the four forms reduce 
to 0. 

In the preceding discussion we have made 
p^>q, p'<q, and p^ = q; 
we have also made p and q separately equal to 0, and then 
both equal to at the same time. 

These suppositions embrace every possible hypothesis that can 
be made upon p and q. i 

1I8» The results deduced in article 117 might have been ob- 
tained by a discussion of the four forms themselves, instead of 
their roots, making use of the principles demonstrated in arti- 
cle 116. 

In the first form the product of the two roots is equal to 

— q, hence the roots must have contrary signs ; their sum is 

— 2p, hence the negative root is numerically the greater. 

In the second form the product of the roots is equal to — y 
and their sum equal to + 2p ; hence, their signs are unlike, 
and the positive root is the greater. 

In the tliird form the product of the roots is equal to -\- q'y 
hence, their signs are alike, and their sum being equal to — 2j3, 
they are both negative. 

In the fourth form the product of the roots is equal to + y, 
and their sum is equal to -\- 2p ; hence, their signs are alike 
and both positive. 

If ^ = 0, the sum of the roots must be equal to ; or the 
roots must be equal with contrary signs. 



CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 161 

If q = 0, the product of the roots is equal to ; hence, one 
of the roots must be 0, and the other will be equal to the co- 
efficient of the first power of the unknown quantity, taken with 
a contrary sign. 

If ^ = and q = 0, the sum of the roots must be equal 
to 0, and their product must be equal to ; hence, the roots 
themselves must both be 0. 

119, There is a singular case, sometimes met with in the- 
discussion of problems, giving rise to equations of the second 
degree, which needs explanation. 

To discuss it, take the equation 

ax"^ -\- bx = c, 



which gives x = ^r . 

If, now, we suppose a = 0, the expression for the value of 

X becomes 

r 



b±b 

— , whence. 



'' = T' 





26 



= 00 





But the supposition a = 0, reduces the given equation to- 
bx = c, which is an equation of the j^rst degree. 

The roots, found above, however, admit of interpretation. 

The first one reduces to the form — in consequence of the 

existence of a factor, in both numerator and denominator, which 
factor becomes for the particular supposition. To deduce the 
true value of the root, in this case, take 

_ — 6 + V^^ + 4ac 

and multiply both terms of the fi-action by — 6 — ^ b"^ -\- ^acy 
after striking out the common factor —2a we shall have 

2c 



b+^b'^-\-^ 
11 



162 ELEMENTS OF ALGEBRA. LCHAP. VI. 

ill which, if we make a = 0, the value of x reduces to -r-^ 

the same value that we should obtain by solving the simple 
equation hx = c. 

The other root oo, is the value towards which the expression, 
for the second value of x^ continaally approaches as a is made 
smaller and smaller. It indicates that the equation, under the 
supposition, admits of but one root in finite terms. This should 
be the case, since the equation then becomes of the first degree. 

120. The discussion of the following problem presents most 
of the circumstances usually met with in problems giving rise 
to equations of the second degree. In the solution of this 
problem, we employ the following principle of optics, viz. : — 

The intensity of a light at any given distance^ is equal to its 
hitensity at the distance 1, divided by the square of that distance. 

Problem of the Lights. 



C" A C B C 

121» Find upon the line which joins two lights, A and J5, of 
different intensities, the point which is equally illuminated by 
the lights. 

Let A be assumed as the origin of distances, and regard all 
distances measured from A to the right as positive. 

Let c represent the distance AB, between the two lights ; 
a the intensity of the light A at the distance 1, and h, the in- 
tensity of the light B at the distance 1. 

Denote the distance AC^ from A to the point of equal illu- 
mination, by x ; then will the distance from B to the same 
point be denoted by c — x. 

From the principle assumed in the last article, the intensity 
of the light A^ at the distance 1, being a, its intensity at the 

distances 2, 3, 4, &;c., will be — , -— , — , &c. ; hence, at thn 
.distance x it will be expressed oy — . 



CHAP. VI.] EQUATIONS OF THE SECOND DEGEEE. 163 

In like manner, the intensity of B at the distance c — x, is 
but, by the conditions of the problem, these two 



(c-xf 

intensities are equal to each other, and therefore we have the 

equation 



a 



x' (c - xf ' 
which can be put under the form 



C — CC ± -i/ft" 

nenee, = '^-— \ whence 



C -y/ll 



■ (1). 



- (2). 



Since both of these values of x are always real, we conclude 
lliat there wdll be two points of equal illumination on the line 
A B^ or on the line produced. Indeed, it is plain that there 
should be, not only a point of equal illumination between the 
lights, but also one on the prolongation of the line joining the 
lights and on the side of the lesser one. 

To discuss these two values of x. 

First, suppose a ^ b. 

The first value of x is positive; and since 

y« <i 

this value will be less than c, and consequently, the first point (7, 
will be situated between the points A and B. We see, moreover, 
that the point will be nearer B than A ; for, since a > 6, we 
have 

^-\-y/a or, 2ya>(y^+y^, whence 

\/^ ^ 1 , ^1 <^ V^ ^ ** 

— ■=r^ 7= > — ; and consequently, — =^ ~ > — . 



164 ELEMENTS OF ALGEBRA. LCHAP. VL 

The second value of x is also positive; but since 

it will be greater than c; and consequently, the second point 
wiL be at some point C\ on the prolongation of AB^ and at 
llie right of the two lights. 

This is as it should be; for, since the light at A is most 
intense, the point of equal illumination, between the lights, ought 
to be nearest the light B\ and also, the point on the prolonga- 
tion of AB ought to be on the side of the lesser light B. 

Second^ suppose a <i b. 
The first value of x is positive ; and since 

this value of x will be less than c; consequently, the first point 
will fall at some point C, to the right of A, and between A 
and B. 



C" A C B C 

We see, moreover, that it will be nearer A than B\ for, 
since a <^ b, we have 

Va+yT>2ya', and consequently, 1.'^ ^ /- < """♦ 

y/a^^b 2 

The second value of x is essentially negative, since the nume- 
rator is positive, and the denominator essentially negative.* 

We have agreed to 4i,consider distances from A to the right 
positive; hence, in accordance with the rule already established 
for interpreting negative results, the second point of equal illu- 
mination will be found at C'\ somewhere to the left of A, 

This is as it should be, since, under the supposition, the light 
at B is most intense; hence, the point of equal illumination, 
between the two lights, should be nearest A^ and the point in 
the prolongation of AB^ should be on the side nearest the 
feebler light A, 



CHAP. VI. J EQUATIONS OF THE SECOND DEGREE. 165 

Third^ suppose a = h^ and c > 0. 

■%. /» 

The firs* value of x is then positive, and equal to — hence, 

the first point is midway between the two lights. 

The second value of x becomes = oo , a result which in- 

dicates that there is no other point of illumination at a finite 
distance from A. 

This interpretation is evidently correct; for, under the suppo- 
sition made, the lights are equally intense, and consequently, the 
point midway between them ought to be equally illuminated. 
It is also plain, that there can be no other point on the line 
which will enjoy that property. 

Fourth^ suppose h ^=.a and c = 0. 
The first value of x becomes, — — = 0, hence the first point 
is at -4. 

The second value of x becomes, — , a result which indicates 

that there are an infinite number of other points which are 
equally illuminated. 

These conclusions are confirmed by a consideration of the con- 
ditions of the problem. Under this supposition, the lights are 
equal in intensity, and coincide with each other at the point A, 
That point ought then to be equally illuminated by the lights, 
as ought, also, every other point of the line on whi'ih the lights 
are placed. 

Fifths suppose a > 6, or a <ib^ and c = 0. 
.Under these suppositions, both values of x reduce to 0, which 
shows that both points of equal illumination coincide with the 
point A, 

This is evidently the case, for, since a is not equal to 5, 
and the lights coincide at A^ it is plain that no other point than 
A can be equally illuminat»ed by them. 

The preceding discussion presents a striking example of the 
precision with which the algebraic analysis respond* to all the 
relations which exist between the quant* ties that enter a problem. 



166 ELEMENTS OF ALGEBRA. [CHAP. YL 



EXAMPLES INVOLVING RADICALS OF THE SECOND DEGREE. 

1. Given, r. -{-/a^ j^ x^ = — , to find the values of i?. 

By reducing to entire terms, we have, ^^^H 

x^aP' + a:2 + a2 + x^ = 2a?, 
oj transposing, ary^a^ + x^ = a^ — x^, 

and by squaring both members, a^x^ -{- x* = a^ — 2a'^x^ + x*^ 
whence, Sa^x^ = a*, 

and, x = ±y^—. 

fa? /o^ 

2. Given, \/— + ^^ — \/~ "~ ^^ = ^? ^^ ^^^^ ^^^ values of x, 

fa? fa? 

By transposing, W — + b^ =\/ — — ^^ + ^ ; 



a? Q? I o? 

squaring both members, -g + ^^ = -2 — ^^ + 2i W -— — 6^ -j- 6' ; 

X X y X 

fcfi fa? 

whence, b^ z= 2b \/ — — b^, and b = 2\/— — b^] 

squaring both members, b^ = —^ — W- \ 

and hence, x^ = — , and a; = ± 



562 j^5 



d /a? x"^ X 

8. Given, f- \/ — = 1. to find the values of x. 

X \ x^ b 



A}is. x= ± V 2ab — 6'. 

4. Given, \/^ -\- V \ — ^^ — = b'^K — ^- — , to nnd ihe 

V a y X -\- a y X -\- a^ 

values of ic. , a 

Ans. X = 



(4 HF IV 



CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 167 

a — -y/ a? — X' 
a +^ CL^ — «^ 



ft ^ / /j2 /^3 

6. Given, ^ = 5, to find the values of x. 



Ans. x= ±L -T^^-T* 
1 +0 



<5. Given, ^ — — • , to find the values of x. 

^/ X — J X —a X — a 

_ Ans. X = -T——r~' 
, 1 ± 2/^ 



X 

Ans. X = ±1 2 J ah — b\ 



I X 

7. Given, ^ "^ ' *" + ^^^—^ ^ — V ~~' *^ ^^^ ^^ values of «. 



8. Given, — ■ = 5, to find the values of x. 

a-\- X 

±a(l ± -/26 - 62) 

^26 - 62 

(y Trinomial Equations. 

122. A trinomial equation is one which involves only terms 
containing two different powers of the unknown quantity and a 
known term or terms. 

123« Every trinomial equation can be reduced to the forni 

x'^ + ^px^ = q (1), 

in which m and n are positive whole numbers, and p and q 
known quantities, by means of a rule entirely similar to that 
given in article 111. 

If we suppose m = 2 and w = 1, equation (1) becomes 
x"^ 4- 2^:r =1 q^ 
a trinomial equation of tho second degree. 

124. The solution of trinomial equations of the second degree, 
has already been explained. The methods, there explained, are, 
with some slight modilcations, applicable to all trinomidl equa- 
tions in which m = 2;i, that is, to all equations of the form 
^2n _^ ^px^ = q. 



168 ELEMENTS OF ALGEBRA. [CHAP. VL 

To demonstrate a rule for the solution of equations of this 
form, let us place 

a:" =: ?/ ; whence, r^ " =2/^. 
Those values of x^ and a;^'*, being substituted in the given 
equation, reduce it to 

y2 _j_ 2py = q, 



whence, y — — jp ± V g + />2, 

or, x-^ — —jp zhy/^ + p^. 

Now, the n*'^ root, of the first member, is x (Art. 18), and 
although we have not yet explained how to extract the ti** 
root of an algebraic quantity, we may indicate the n^^ root of 
the second member. Hence, (axiom 6), 



I 

Hence, (^axiom tjj, i 



Hence, to solve a trinomial equation which can be reduced 
to the form a;2« -f- 2^2;" = q^ we have the following 

RULE. 

Reduce the equation to the form of x'^'^ + ^px^ = 3^ / i^^ valuer 
of the unknown quantity will then be found by extracting the 
71^^ root of half the co-efficient of the lowest power of the un- 
known quantity with its sign changed^ plus or minus the square 
root of the second member increased by the square of half the 
co-efficient of the lowest power of the unknown quantity. 

If n =z2, the roots of the equation are of the form 



X= ±yj -p ±: ^q+p^ 



"We see that the unknown quantity has four values, since each 
of the signs + and — , which affect the first radical can be 
combined, in successicn, with each of the signs which affect the 
second ; but these values, taken tcvo and two, are numerically equals 
and have contrary signs. 




CHAP. VL] TRINOMIAL EQUATIONS. 169 



EXAMPLES. 



1, Take the equation 

a;4_25a;2 = --144. 
Tliis being of the required form, we have by app ication of 
the rule, ^ , 

/^. — , e> tf 



^^i 



/25 7" /, ' 

whence, ic = zb W — - ± — ; ^l' 

hence, the four roots are +4, — 4, +3, and — 3. 

2. As a second example, take the equation 

X* - 7a;2 = 8. 
Whence, by the rule. 



hence, the four roots are, 

+ 2y/2; -2/2; +/^n: and -/^=T; 
the last two are imaginary. 
S, X*- {2bc + 4a2) .t;2 = - Pc^ 

' * . ' ^ . ,:' ■.' '^'' I 

" ■ - ' ^ iUtk,^j*^ ^Ans. x= ±x/bc-{-2a^ dz 2a^bc + a^. 
4. 2a; - 7^1 = 99. Ans. x = 81, x = ^. 



5. ^_6,i4 4-4^'=0. Ans, x=±./I±V^^^^fJ:Z, 
^ ^ ^ 2bd ' 

125. The solution of trinomial equations of the fourth degree 

requires the exfraction of the square root of expressions of the 

form of a zb V^" in whicji a and b we positive or negative, 

numerical or algebraic. The expression v/ a —x^b can some- 

times be reduced to the form of a' ±i Jl/ or\to the form 
J a'' ±i -J ^ ' j ^^'i when such transformation is possible, it w 



170 ELEMENTS OF ALGEBRA. LCHAF. VL 

advaiitageou'j to effect it, since, in tiiis case, we have only to 
extract two simple square roots ; whereas, the expression 



\/« 



requires the extraction of the square root of the square root. 

To deduce formulas for making the required transformation, 
let us assume 

p + q = sj^+/h]. . . - (1), 



= \A^/^ ■ (2); 



p-q 

in which p and q are arbitrary quantities. 

It is now required to find such values for p and q as will 
satisfy equations (1) and (2). 

By squarmg both members of equations (1) and (2), we have 
^2 + 2p^ + 9' = «+V^- ' " (^)' 
p2_2pqJ^f=,a-^T. - . (4). 
Adding equations (3) and (4), member to member, we get 

p"^ -{- q"^ — a (5). 

Multiplying (1) and (2), member by member, we have, 
p^ — q^ =:J a? — h. 

Let us now represent J o?' — h by c. Substituting in the 
last equation, 

p-^-q^^c (6). 

From (5) and (6) we readily deduce, 

P=^\l^ and g = ±y/^., 

these values sibstituted for p and g, in equations (1) and (2), 
give 



vW^=-\/T-v/^ 



— c 



Chap, vi.] trinomial equations. 171 

u6IlC6 

and ^^^=-(^/^-v^) - - («)• 

Now, if a? — h is a perfect square, its square root, c, will * 
bo a rational quantity, and the application of one 0/ the for- 
mulas (7) or (8) will reduce the given expression to the re- 
quired form. If a^ — h is not a perfect square, the application 
of the formulas will not simplify the given expression, for, we 
shall still have to extract the square root of a square root. 

Therefore, in general, this transformation is not used, unless 
a^ —b is a perfect square. 



EXAMPLES. 



1. Keduce </ 94 + 42 y^5 =\/ ^^ + ^8820, to its simplest 
form. We have, ' a = 94, 6 rr 8820, 

whence, c =^a^ — b =y/8836 — 8820 = 4, 

a rational quantity ; formula (7) is therefore applicable to this 
case, and we have 

or, reducing, = ± (.V49 +y^45) ; 



hence, a/94 + 42^5 = ± ( 7 + 3^5). 

This may be verified; for, 

(7 -I- 3^)2 = 49 + 45 + 42^ = 94 + 42y^. 

2. Reduce y/ ^P + ^^^ ~ '^m.^prip + m^, to its simplest 
form, y e ha-^e 

a — njp A- 2m2, and b = 4m2(n^ 4- m2), 
a* - '> ~ yi^jf"^ and c — y/ o? — l> = np ; 



172 ELEMENTS OF ALGEBRA. [CHAP. VL 

and therefore, formula (7) is applicable. It gives, 

/ /np -j~ 2m2 + wp fnp + 2m? — np\ 

"^W 2 V 2 > 

an'3, reducing, ± {^ np -\- ni^ — iii). 

3. Reduce to its simplest form. 



y/l6 + 30^-1 ^U 16 - 3 0, 
By applying the formulas, we find 



and 



-y/l6 -f- 30,/^ .= ^.di??^!^!' 
-y/ie - SOy^ = 5 - ^y/^^ : 
-1/16 + 30^"^^ +a/i6 - SOy^^i 



hence, W 1^ + ^^V " 1 +\/ l^^ " ^^V ~^ = ^^' 

This example sho^vs that the transformation is applicable to 
imaginary expressions. 

4. Reduce to its simplest form, 

y/28 + 10^3. Ans. 5 +^3. 

5. Reduce to its simplest form. 



6. Reduce to its simplest form, 

\/bc + 2h^bc-lP- - Uhc — 2b^bc - 62. 

7. Reduce to its simplest form, 



yj ab -f 4c2 - c/2 - 2y/ 4a6c2 - abd'^. 

Ans. yfab -- /4c' — d^ 



CHAP. 71.] EQUATIONS OF THE SECOND DEGREE. 173 

Equations of the Second Degree involving two or more unhnovm 

quantities. 

126t Every equation of the second degree, containing two 
unknown quantities, is of the general form 

ay2 ^ i:cy + cx^ + dy ■\- fx + y = ; 
or a particular case of that form. For, this equation contains 
terms involving the squares of both unknown quantities, theii 
product, their first powers, and a known term. 

In order to discuss, generally, equations of the second degree 
involving two unknown quantities, let us take the two equations 
jf the most general form 

ay'^ -\-hxy -{■cx'^-^dy +fx-\-g =0, 
and a'y"^ + h'xy + c'x"^ + d'y -\-f'x -}- ^' = 0. 

Arranging them with reference to x, they become 
cx"^ -{-{by -]rf)x'\-a7/-\- dy + g =0, 
cV + {b'y +f') X + aY J^d'y + g' = ()', 
from which we may eliminate a;^, after having made its co-effi- 
cient the same in both equations. 

By multiplying both members of the first equation by c', and 
both members of the second by c, they become, 

cc'x^ -[.{hy^-f)c'x-^{ay^ + dy + gy = 0, 
cc'x' + {h'y +/)c X + (ay + a'y + g'y = 0. 

Subtracting one from the other, member from member, we have 

[{be' — cb')y -\-fc' — c/'~\x -\- (ac — ca')y'^ -f- {dC — cd')y -f gc' 

- ^9' = 0, 
which gives 

_ (ca' — ac')y'^ + {cd' — dc')y -f eg' — gc' 
* - {be' - cb')y -\-fc' - cf • 

This value being substituted for x in one of the proposed 
equations, will give a final equation, involving only y. 

But without effecting the substitution, which would lead to a 
very complicated result, it is easy to perceive that the final 
equation involving y, will be of the fourth degree. For, the 



174 ELEMENTS OF ALGEBRA. [CHAF. VI, 

numerator of the value of x being of the form 

its square will be of the fourth degree, ani this square forms 
one of the parts in the result cf the substitution. 

Therefore, in general, the solution of two equations of the secona 
degree, involving two unknown quantities, depends upon that of an 
equation of the fourth degree, involving one unknown quantity. 

127. Since we have not yet explained the manner of solving 
equations of the fourth degree, it follows that we cannot, as 
yet, solve the general case of two equations of the second 
degree involving two unknown quantities. There are, however^ 
some particular cases that admit of solution, by the application 
of the rules already demonstrated. 

First. We can always solve two equations containing two 
unknown quantities, when one of the equations is of the second 
degree, and the other of the first. 

For, we can find the value of one of the unknown qua^i 
titles in terms of the other aiid known quantities, from the 
latter equation, and by substituting this in the former, we shall 
have a single equation of the second degree containing but one 
unknown quantity, which can be solved. 

Thus, if we have the two equations 

.r2 + 2y2 = 22 - - - (1), 

2x - y = \ . . . . (2), 
we can find from equation (2), 

^==1 + 1; whence, ^^ ^\±^t . 
and by substituting this expression for x"^ in equation (1), we find 

2 



whence we get the values of y : that is, 

y = Z and y = - — ; 

and by substituting in equation (2) we find, 
ar = 2 and x = — --. 

9 



cffAP. VI.] EQUATIONS OF THE SECOND DEGKEE. 175 

Second. We can always solve two equations of the second 
degree containing two unknown quantities when they are boCh 
homogeneous with respect to these quantities. 

For, we can substitute ' for one of the unknown quantiti»:«s, 
an auxiliary unknown quantity multiplied into the second un- 
known quantity, and by combining the two resulting equations 
we can find an equation of the second degree, from which the 
value of the auxiliary unknown quantity may be determined, 
and thence the values of the required quantities can easily be 
found. 

Take, for example, the equations 

x^-^ xy — y'^ — b - - - (1), 
3a;2 _ 2^y — 2y2 = 6 - - - (2). 

Substitute for y, px^ p being unkno-svn, the given equat. /lis 
become 

3a:2 - 2i?.'c2 - 2i?2a:2 :^ 6 . . . (4), 
Finding the values of x"^ in terms of p, from equation? ^S) 
and (4), and placing them equal to each other, we deduce 
5 6 





l + p-p'^ 3-2^-2^2 


or reducing, 


9 


whence. 


^ A ^ 

jp = — , and P=--^- 



Considering the positive value of p^ we have, by substituting 
it in equation (3), 

or, a;2 = 4 ; 

whence, x = 2 and a; = — 2 : 

and since y =. px we have y = 1 and y = — \, 

Third There are certain other cases which admit of solution, 
but for wnich no fixed rule can be given. 

We shall illustrate the manner of treating these cases, ^j 
the solution of the following 



17^ ELEMENTS OF ALGEBRA. [CHAP. VL 



\. Given, -^ = 48, 



EXAMPLES. 



X 

y \ to find the values of x and y. 

^ = 24, 

Dividing the first by the second, member by member, we have 

■y/ X f 

^ = 2, or Vy = 2 ; whence y = 4 ; 

and by substituting in the second equation, we get 
y/^ = 6, and x = 36. 

2. Given, x ^~ /xy-\- y = 19, ) ^ , , , - , 

^ „ >• to find the values of x and y, 

a;2 _|. a;y 4- 2^2 _ 133^ j y 

Dividing the second by the first, member by member, we 
have 

^ — V^ 4- y = 7. 

But, a; + V^ + y = 19 : 

adding these, member to member, and dividing by 2, we find 



;.^^2^;>^ ,;^V a; + y = 13. 


which substituted ] 


m the first equation, gives. 


^xy = 6, 


36 

or xy = 36, and x = — . 


Substituting this 


expression for x, in the preceding equation, 


we get. 


?+,-.=, 


or, 


y2 _ 13y = _ 36 ; 


whence, y = 


= 2 -V '^^+4=2^2- 


and finally, 


y = 9, or 2/ = 4; 


and since 


x-\-y = lS, 




a: = 4, or x = 9. 



CHAP. VI.J EQUATIONS OF THE SECOND DEGREE. 177 

3. Find the values of x and y, in the equations 
a;2 4- 3^ + y = 73 — 2a;y 
2/2 + 32/ + a; = 44. 
, By transposition, the first equation becomes, 

ic2 + 2a;2/ -I- 3a; + 2/ = 73 ; 
to which, if the second be added, member to memberj tuere 
results, 

a;2 + 2^yj-j^2 + 4a; + 4y = (a; -^ yf + 4 (a; -j- y) = 117. 
If, now, in the equation /^. j^/ ^ ,,/ ^ ^ i "^-^^^ '^ 

(a; + y)2 + 4 (a; + y) = 117, \.-\^^ 
we regard a; + y as a single unknown quantity, we shall Lave 

a; + 2^=-2±yil7 + 4; 
hence, a; + 2/= — 2+11 =9, 

and a; + y = - 2 — 11 = — 13 ; 

whence, a; = 9 — ?/, and a; = -- 13 — 2^. 

Substituting these values of x in the second equation, we have 

2/2 + 2y = 35, for a; = 9 -— 2/> 
and 2/2 _|_ 2?/ = 57, for a; = — 13 — y. 

The first equation gives, 

y = 5, and y = — 7, 
and the second, 

y= — 1+^58^ and y= — 1—^58. 
Ilie corresponding values of a;, are 

a; = 4, a; = 16 ; 
a; =-12-^58, and a; =-12+^58. 

4. Find the values of x and y, in the equations 
a;2y2 -I- a;y2 + a^y = 600 — (y + 2) a; V 
a; + y2 = 14 — y. 
From the first equation, we have 

a:2y2 + (y2 + 2y) a;2y2 -f xy^ + xy = 600, 
or, a;2y2(l+2/' + 2y)+a;y(l+y) =600, 

or, agam, a;2y2 (1 + y)i + a;y (1 + y) = 600 ; 

12 



178 ELEMENTS OF ALGEBRA. [CHAP. VL 

which is of the form of an equation of the second degree, r©. 
garding xy (1 + v) as the unknown quantity. Hence, 

-<.; :.j, (1 + y) = - J ± yeoo+l = - i ± sj~ ; '" ' - 

and if we discuss only the roots which belong to the + value 

of the radical, we have 

1 49 

:ry(l+y)=:-- + ^ = 24; 

24 
and hence, x = — ; — - . 

Substituting this value for x in the second equation, we have 

(y' + y)'-14(y2 + y) = -24; 
whence, y"^ -\- y = 12, and y^ -\- y = 2. 

From the first equation, we have 

y=--^±-j = S, or -4; 

and the corresponding values of x, from the equation 

24 

y^ -{- y 

From the second equation, we have 

y = 1, and 2/ = — 2; 
which gives a; = 12. 

6. Given, x'^y + xy^ = 6, and x^y^ + x^y^ = 12, to find the 

values of x and y. ( a; = 2 or 1 

Ans. i 

or 2. 



(x = 2 



« r" ( a;2 + a; + y = 18 — y2 i ^ fi^^j ^he values of 

D. ijriven, < V 

i xy = Q ) X and y. 

Ans. \' = ^^ ^^ ^' ^^ -3±v^ 
iy = 2, or 3 ; or - 3 + /SL 

Problems giving rise to Equations of the Second Degree con 
taining two or more unknown quantities. 

1. Find two numbers such, that the sum of the respect' v, 
products of the first multiplied by a, and the second multiplied 
by 6, shall be equal to 2s; and the product of the one by 
the other equal to p. 



CHAP. VI.] EQUATIONS OF THE SECOND DEGREE. 179 

Let X and y denote the required numbers, and we have 
<Mf + 6y = 25, 



and 


^y^v- 


From the first 






2s — ax 

y - A 5 



whence, by substituting in the second, and reducing, 

ax^ — 2.SX = — bp. 

Therefore, x z= — dz —J s^ — abp, 

a a V 

and consequently, y =— ^ —y/^--abp. 

Let a = 5 = 1 ; the values of x, and y, then reduce to 
X = s ±^s^ —p, and y = s ^^^ s^ —p; 

whence we see that, under this supposition, the two values 
of X are equal to those of y, taken in an inverse order ; which 
shows, that if 

s + ^ s^ — p represents the value of x, s — ,Vs^ — p 

will represent the corresponding value of y, and conversely. 

This relation i^ explained by observing that, under the last 
supposition, 'h'? / /on equations become 

X -\- y — 2s, and xy = p; 

and 'he '^i^.^ticn is then reduced to finding two numbers of which 
the sur,t is 2.?, and their product p ; or in other words, to divide 
a number 2s, into two such parts, that their product may be equal 
to a given number p, 

2. To find four numbers, such that the sum of the first and 
fourth shall be equal to 2^, the sum of the second and third 
equal to 2s', the sum of their squares equal to 4.c^, and the 
product of the first and fourth equal to the product of the 
Recond and third. 



180 ELEMENTS OF ALGEBKA. [CHAP. VI, 

Let Wj ir, y, aai 2, denote the numbers, respectively. Then, 
from the conditions of the problem, we shall have 

u -{- z =2s 1st condition j 

x-{-y z=z2s' 2d " 

MH-«^ + y=^ + ^^ = 4c2 3d " 

uz = xy 4th " 

At first sight, it may appear difficult to find the values of 
the unknown quantities, but by the aid of an auxiliary unknown 
quantity^ they are easily determined. 

Let p be the unknown product of the 1st and 4th, or 2d 
and 3d ; we shall then have 

«--/s2 -p. 
aiid 

•j y which give, -j ^ 

( ^y=P, ) iy =zs'—^s'^—p. 

Now, by substituting these values of ^^, x, y, z, in the third 
equation of the problem, it becomes 



[u-\-z = 2s,) (u = 

^ [. which give, < 

( UZ=p, ) lzz= 



and 'nj developing and reducing, 

4^2 j^ ^g'2 — 4p = 4c2 . hence, j9 = s^ + 5'^ — c^. 

Substituting this value for p^ in the expressions for -w, x, y, «, 
we find 



w = s+yc2 — 5'2, ja;=5'+^ 



;2 _ ^2^ 



(z = 5 —Jc^ — a'2, ly =S' —Jc^ «^. 

These values evidently satisfy the last equation of the 
problem; for 

W2 = (S +yc2-s'2) (S -yc2-s'2) = S^ - C* -f s'^, 



ary = (s'+^c^^ s 2) (5/ ..^^2-^5 2) = s'2 _ ^2 ^. « 2. 



CHAP. VI.] EQUATIONS OF THE SECOND DEGEEE. 181 

Remark. — Tliis problem shows how much the introduction 
of an unknown auxiliary often facilitates the determination of 
the principal unknown quantities. There are other problems 
of the same kind, which lead to equations of a degree supe- 
rior to the second, and yet they may be resolved by the aid of 
equations of the first and second degrees, by introducing unknown 
auxiliaries. 

3. Given the sum of two numbers equal to a, and the sum 
of their cubes equal to c, to find the numbers 

Ix -\- y z=za 
By the conditions I 



y^ =c. 



Putting X z= s -{- Zj , and y = s — z^ we have a = 2s-, 

3 = .s3 + Ss^z + Ssz^ + z^ 
y^ =s^ — Ss^z 4- 352;2 _ ^3 . 



( X^ = .S3 + 
^^^ 1 3 3 

iy^ =s^ — 
hence, by addition, x^ -{-y^ = 2s^ -f Qsz"^ 



, c-2s^ /c-2s' 
whence, z^ = — , and g = ±\/ — 

05 V 05 



"'-\/^^"67^' ^""^ y -«+>/- 



'c 2s^ 

or, X =z s ±K / — :: . and y=±s^\/ — ; 

and by substituting for s its value. 



i-V^)~iW- 



4c — a^ 
12a ' 



^"^ ^=^W(h^)=^W- 



4c — 

~l2a 



4. The sum of the squares of two numbers is expressed by 

a, and the difference of their squares by b : what are the 

numbers] /;rTl> 17^ h 

Ans 



^Vv 



2 ' V 2 

5. What three numbers are they, which, multiplied two and 
two, and each product divide I by the third number, give the 
quotients, a, 6, c1 

Ans. ^ib, y^ yp^. 



182 ^^Av/«: ELEMENTS OF ALGEBRA. I CHAP. VL 

6. The sum of two numbers is 8, and the sum of their 
cubes is 152 : what are the numbers % Ans. 3 and 5. > 

7. Find two numbers, whose difference added to the differ- 
yT ence of their squares is 150, and whose sum added to the 
/ sum of their squares, is 330. Ans. 9 and 15. 

\ 8. There are two numbers whose difference is 15, and half 
( their product is equal to the cube of the lesser number : what 
are the numbers? Ans. 3 and 18.^ 

T9. What two numbers are those whose sum multiplied by 
the greater, is equal to 77 ; and whose difference, multiplied 
by the lesser, is equal to 12 1 

Ans. 4 and 7, or | V2" and y V2^ 

10. Di\dde 100 into two such parts, that the sum of their 
square roots may be 14. Ans. 64 and 36. 

11. It is required to divide the number 24 into two such 
parts, that their product may be equal to 35 times their differ- 
ence. Ans. 10 and 14. 



^ 



12. What two numbers are they, whose product is 255, and 
the sum of whose squares is 5141 Ans. 15 and 17. 

13. There is a number expressed by two digits, which, when 
divided by the sum of the digits, gives a quotient greater by 
2 than the first digit ; but if the digits be inverted, and the 
resulting number be divided by a number greater by 1 than 
the sum of the digits, the quotient will exceed the former 
quotient by 2 : what is the number 1 Ans. 24. 

14. A regiment, in garrison, consisting of a certain number of 
companies, receives orders to send 216 men on duty, each com- 
pany to furnish an equal number. Before the order was exe- 
cuted, three of the companies were sent on another service, 
and it was thcr. found t'aat each company that remained would 
have to send 12 men additional, in order to make up the com. 
plement, 216. How many companies were in the regiment, and 
what number of men did each of the remaining companies send 

Ans. 9 companies : each that remained sent 36 men. 



CHAP VI. J EQUATIONS OF THE SECOND DEGKEE. 183 

15. Find three numbers such, that their sum shall be 14, the 
sum of their squares equal to 84, and the product of the first 
and third equal to the square of the second. 

Ans. 2, 4 and 8. ' 

16. It is required to find a number, expressed by three 
digits, such, that the sum of the squares of the digits shall 
be 104; the square of the middle digit to exceed twice the 
product of the other two by 4 ; and if 594 be subtracted from 
the number, the remainder will be expressed by the same 
figures, but w^ith the extreme digits reversed. Ans. 862. 

-"^ 17. A person has three kinds of goods which togetner cost $2302^. 
A pound of each article costs as many -Jj dollars as there are 
pounds in that article : he has one-third more of the second than of 
the first, and 3^ times as much of the third as of the second : How 
many pounds has he of each article ? 

Ans.i 15 of the 1st, 20 of the 2d, 70 of the 3d. 

18. Two merchants each sold the same kind of stufi": the 
second sold 3 yards more of it than the first, and together, 
Uiey received |I5 dollars. The first said to the second, " I 
would have recd>ived 24 dollars for your stufl!'." The other re- 
plied," "And 1/ would have received 12J dollars for yours." 
How many ya^-ds did each of then sell? 

/' , (1st merchant 15 ) (5 

1 ^"'- isd - - - isf "' ia 

19. A wir^ow possessed 13000 dollars, which she divided into 
two parts, v^nd placed them at interest, in such a manner, that 
the income (6 from them were equal. If she had put out the first 
portion at the same rate as the second, she would have drawn 
for this pjart 3G0 dollars interest; and if she had placed the 
second oujt at the same rate as the first, she would have drawn 
for it 4.9^i dol'ars interest. What were the twc rates of interest? 

Ans, 7 and 6 per cent 



CHAPTER VU. 

rORMATION OF POWERS BINOMIAL THEOREM — EXTUACnON »F ROOTS OP 

ANY DEGREE — OF RADICALS. 

128. The solution of equations of the second degree supposes 
the process for extracting the square root to be known. In 
like manner, the solution of equations of the third, fourth, &e., 
degrees, requires that we should know how to extract the third, 
fourth, &c., roots of any numerical or algebraic quantity. 

The power of a number can be obtained by the rules for 
jnultiplication, and this power is subject to a certain law of for- 
mation, which it is necessary to know, in order to deduce the 
root from the power. 

Now, the law of formation of the square of a numerical or 
algebraic quantity, is deduced from the expression for the squar« 
of a binomial (Art. 47) ; so likewise, the law of a power of 
any degree, is deduced from the expression for the same power 
of a binomial. We shall therefore first determii)e the law for 
the formation of any ■power of a binomial. 

129. By taking the binomial x -\- a several times, as a factor, 
the following results are obtained, by the rule for multiplication* 

{x -\- a^ =z X -\- a, 

(x + a)2 =zx^-\- 2ax + a^ 

{x + a)3 = x^-\- Zax^ + ^aH -f a^, 

(a -f a)^ = a;* -f Aiax^ + Ba^^z _|_ 4^353^ _|_ ^\^ 

(j; -f a)5 — x^ 4- Saar* -f lOa^a:^ + X^a^x^ + 5a*ar -f ^a*. 

By examining these powers of a: + a, we readily dis(\over tht 
law according to which the exponents of the powers cj>f a ie 



CHAP. VII. J PERMUTATIONS AND COMBINATIONS. 185 

crease, and those of the powers of a increase, in the successive 
terms. It is not, however, so easy to discover a law for the 
formation of the co-efficients. Newton discovered one, by nneans 
of which a binomial may be raised to any power, without per 
fuiming the multiplications. He did not, however, explain the 
course of reasoning which led him to the discovery ; but the law 
has since been demonstrated in a rigorous manner. Of all the 
known demonstrations of it, the most elementary is that which 
is founded upon the theory of combinations. However, as the 
demonstration is rather complicated, we will, in order to simplify 
it, begin by demonstrating some propositions relative to permu- 
tations and combinations, on which the demonstration of the 
binomial theorem depends. 

(y Permutations^ Arrangements and Oomhinations. 

130« Let it be proposed to determine the whole number oj 
mays in which several letters, a, b, c, c?, &c., can be written, 
one after the other. The result corresponding to each change 
in the position of any one of these letters, is called a per 
mutation. 

Thus, the two letters a and b furnish the two permutations, 

nb and ba, 

cab 

acb 
In like manner, the three letters, a, b, c, furnish abc 

six permutations. | cba 

bcd 

^baa 

Permutations, are the results obtained by writing a certain 

number of letters one after the other, in every possible order, in 

S2ich a manner that all the letters shall enter into each result^ and 

each letter enter but once. 

To determine the number of permutations of which n letters are 
susceptible. 

Two letters, a and 6, evidently give two per- , j ab 

mutations. ( ba 



^ cah 

ach 
abc 
cba 
bca 



186 ELEMENTS OF ALGEBRA. ICHAP. VL 

Therefore, the number of perrmtations of two letters is ex 
pressed by 1x2. 

Take the three letters, a, h, and c. Reserve r c 

either of the letters, as c, and permute the other } ai 

two, giving ( ba 

Now, the third letter c may be placed before aJ, 
between a and b, and at the right of ab; and the 
same for ba : that is, in one of the first 'permuta- 
tions^ the reserved letter c may have three different 
•places^ giving three permutations. And, as the same 
may be shown for each one of the first permutations, 
it follows that the whole number of permutations of 
three letters will be expressed by, 1x2x3. t bae 

If, now, a fourth letter d be introduced, it can have four 
places in each one of the six permutations of three letters : 
hence, the number of permutations of four letters will be ex- 
pressed by, 1x2x3x4. 

In general, let there be n letters, a, 5, c, &;c., and suppose 
the total number of permutations oi n — 1 letters to be known ; 
and let Q denote that number. Now, in each one of the Q per- 
mutations, the reserved letter may have n places, giving n per- 
mutations : hence, when it is so placed in all of them, the 
entire number of permutations will be expressed by Q X n. 

If n rr 5, Q will denote the num.ber of permutations of four 
quantities, or will be equal to 1x2x3x4; hence, the num- 
ber of permutations of five quantities will be expressed by 
1x2x3x4x5. 

If w = 6, we shall have for the number of permutations of 
MX quantities, 1x2x3x4x5x0, and so on. 

ITence, if Y denote the number of permutations of n letters, 
wi shall have 

F= Qxn=z\. 2. 3. 4. . . . [n — VjU'. that is, 

The number of permutations of n letters^ is equal to the con^ 
U mcd product of the nntural numbers from I to n inclusively. 



CHAP. VI.] PERMUTATIONS AND COMBINATIONS. 187 

An^angements. 

131. Suppose we have a number m, of letters a, 6, c, c?, &c 
ft they are written in sets of 2 and 2, or 3 and 3, or 4 and 4 
... in every possible order in each set, such results are called 
arrangements. 

Thus, ah^ ac^ ad, . . . ba, be, bd, . . . ca, cb, cd, . . . are ar- 
'•angements of m letters taken 2 and 2 ; or in sets of 2 each. 

1)1 like manner, abc, abd, . . . bac, bad, . . . acb, acd, . . . are 
irrangements taken in sets of 3. 

Arrangements, are the results obtained by writing a number m 
./ letters, in sets of 2 and 2, 3 and 3, 4 and 4, . . . w and n ; 
\,\e letters in each set having every possible order, and m being 
a'ways greater than n. 

If we suppose m =z n, the arrangements, ♦•Jvken n and n, bo- 
come permutations. 

Having given a number m of letters a, h, c, d, . . . to deter- 
mine ihe total number of arrangements that may be formed of them 
by taking them n in a set. 

Let it be proposed, in the first place, to arrange three letters, 
a, h and c, in sets of two each. 

First, arrange the letters in sets of one each, and 
for each set so formed, there will be two letters 
reserved: the reserved letters for either arrange- 
ment, beiiig those which do not enter it. Thus, with 
reference to a, the reserved letters are b and c ; with reference 
to b, the reserved letters are a and c ; and with reference to c, 
they are a and b. 

Now, to any one of the letters, as a, annex, in • ^" 

succession, the reserved letters b and c: to the 
second arrangement b, annex the reserved letters a < 



ac 

ba 

be 
and c and to the third arrangement, c, annex the 

reserved letters a and b. 



ca 

Since each of the first arrangements gives as many new 
arrangements as there are reserved letters, it follows, that the 



ah 

ac 
ad 
ha 
he 
hd 
c a 
cb 
cd 
da 
db 
dc 



188 ELEMENTS OF ALGEBRA. [CHAP. VIL 

number of arrangements of three letters taken^ iwo in a set, will be 
equal to the number of arrangements of the same letters taken one 
in a set, multiplied by the number of reserved letters. 

Let it be required to form the arrangement of four letters, 
a, 6, c and d, taken three in a set. 

First, arrange the four letters in sets of two : there 
will then be for each arrangement, two reserved let- 
ters. Take one of the sets and write after it, in sue- i 
cession, each of the reserved letters : we shall thus 
form as many sets of three letters each as there are . 
reserved letters ; and these sets differ from each other ^ "* 
by at least the last letter. Take another of the first 
arrangements, and annex, in succession, the reserved 
letters ; we shall again form as many different arrange- 
ments as there are reserved letters. Do the same for 
all of the first arrangements, and it is plain, that the 
whole number of arrangements which will be formed, of four 
letters, taken 3 and 3, will he equal to the number of arrange- 
ments of the same letters, taken two in a set, multiplied by the 
number of reserved letters. 

In general, suppose the total number of anangements of m 
letters, taken n — 1 in a set, to be known, and denote this num- 
ber by P. 

Take any one of these arrangements, and annex to it, in suc- 
cession, each of the reserved letters, of which the number is 
m — {n — \), or m — n -{- 1. It is evident, that we shall thi'.s 
form a number m — n -\- 1 of new arrangements of n letters, 
each differing from the others by the last letter. 

Now, take another of the first arrangements of n — - 1 letters, 
and annex to it, in succession, each of the m — n -\- \ letters 
which do not enter it ; we again obtain a number m — n -\- \ of 
arrangements of n letters, differing from each other, and from 
those obtained as above, by at least one of the n — \ first letters. 
Now, as we may in the same manner, take all the P arrange- 
ments of the m letters, taken n — \ in a set, and annex to thero, 



CHAP. VII.] PERMUTATIONS AND COMBINATIONS. 189 

in succession, each of the 7n — n -{- I other letters, it follows 
that the total number of arrangements of m letters, taken n ic 
a set, is expressed by 

F{m - « + 1). 

To apply this, in determining the number of arrangements of 
f« letters, taken 2 and 2, 3 and 3, 4 and 4, or 5 and 5 in a 
set, make n = 2 ; whence, m — w+l = m — 1; P in this 
case, will express the total number of arrangements, taken 2 — 1 
and 2 — 1, or 1 and 1 ; and is consequently equal to m; there- 
fore, the expression 

F{m — n -{- I) becomes m(m — 1). 

Let 71 = 3 ; whence, m — n-^l^zm — 2; P will then ex- 
press the number of arrangements taken 2 and 2, and is equal 
to m[m, — \)\ therefore, the expression becomes 

m{m — V)[m — 2). 

Again, take n = 4 : whence, m— n + l=m — 3: P will ex 
press the number of arrangements taken 3 and 3, and therefore 
the expression becomes 

m{m — \){m — 2)(m — 3), and so on. 

Hence, if we denote the number of arrangements of m let* 
ters, taken n m b, set by X, we shall have, 

X z= P(m — n •\-\) z=m(m —1) (wi — 2) . . (m ~ ?i + 1) ; that is, 

The number of arrangements of m letters, taken n in a set, u 
equal to the continued product of the natural numbers from m 
down to m — n -\- 1, inclusively. 

If in the preceding formula m be made equal to n, the ar 
rangements become permutations, and the formula reduces to 

X=n{n-l){n-.2) ....2.1; 

or, by reversing the order of the factors, and writing Y for X, 

r=l . 2 . 3 . . . . (7i-l)7i; 

the same formula as deduced in the last article. 



\ 



190 ELEMENTS OF ALGEBRA. [OHAP. YH 

Combinations. 

132i When the letters are disposed, as in the arrangements, 
2 and 2, 3 and 3, 4 and 4, &c., and it is required that any 
two of the results, thus formed, shall differ by at least one 
letter, the products of the letters will be different. In this case, 
the results are called combinations. 

Thus rf6, ac, be, . . . ad, bd, . . . are combinations of the let- 
ters a, 6, c, and d, &c., taken 2 and 2. 

In like manner, abc, abd, . . . acd, bed, . . . are combinations 
of the letters taken 3 and 3 : hence. 

Combinations, are arrangements in which any two will differ 
from each other by at least one of the letters which enter them. 

To determine the total number of different combinations thai 
can be formed of m letters, taken n in a set. 

Let X denote the total number of arrangements that can be- 
formed of m letters, taken n and n : Y the number of per 
mutations of n letters, and Z the total number of different 
combinations taken n and n. 

It is evident, that all the possible arrangements of m letters 

taken n in a set, can be obtained, by subjecting the n letters 

of each of the Z combinations, to all the permutations of which 

these letters are susceptible. Now, a single combination of n 

letters gives, by hypothesis, Y permutations or arrangements • 

therefore Z combinations will give Y X Z arrangements ; and 

as X denotes the total number of arrangements, it follows that 

X 
X=Yx Z: whence, Z = --. 

But we have (Art. 130), 

Y= Qxn = \ . 2 . ^ . . , , n, 
and (Art. 131), 
X =1 P {m ~ n ■\- \) = m(m - \) [m -2) . . . . (tti - n f 1) ; 
therefore, 

P(m -71-1- 1)_ m(m — \){m — 2) . . ■ . {m —n-^-l) 

~ Q'yTn ~ 1.2.3 n ' 

that is. 



CHAP. VII. J BINOMIAL THEOREM. 191 

The number cf combinations of m letters taken n in a set^ 
is equal to the continued product of the natural numbers from 
m down to m — n -\- \ inclusively/, divided by the continued 
product of the natural numbers from 1 to n inclusively. 

133. If Z denote the number of combinations of the m let- 
ters taken n in a set, we have just seen that 

m ( m — 1) (m — 2) . . . . {ni — n + 1 ) 

^ - rT2T3 n ^^^- 

If Z' denote the number of combinations of m letters taken 
(m — n) in a set, we can find an expression for Z' by chang- 
ing n into m — n in the second member of the above formula ; 
whence 

_ 7^(m-l)(m-2) 0^+l) .,. 

1.2.3 (m -n) ^ ^' 

If, now, we divide equation (1) by (2), member by member, 
and arrange the factors of both terms of the quotient, we 
shall have 

Z _ 1 . 2 . 3 . . . . (m — w) X (m — w -f- 1) . . . (m — \)m 
'Z' ~ rT2 . 3 . . . . ~ ?i X (n -f 1) (m — l)m' 

The numerator and denominator of the second member are 
equal to each other, since each contains the factors, 1, 2, 3, 
&c., to m; hence, 

— = 1, or Z :=^ Z' \ therefore, 

The number of combinations of m letters^ taken n in a set, is 
equal to the number of combinations of m letters, taken m — n in 
a set. 

'"^ Binomial Theorem. f"^' 

134. The object of this theorem is to show how to ficd any 
power of a binomial, without going through the process of con 
turned multiplication. 

135. The algebraic equation which indicates the law of for- 
mation of any powder of a biromial, is called the Binorrdat 
Formula. 



192 



ELEMENTS OF ALGEBRA. 



[CHAP. VII. 



In order to discover this law for the mth power of the bino- 
mial X -\- a, let us observe the law for the formation of the 
product of several binomial factors, ae + a, x -\- b^ x -\- c, x -^-d 

. . of which the first term is the same in all, and the second 
terms different. 



X •}- a 
X + b 



1st product - x^ 



+ c 



+ ab 



2d 



x"* -f- a 
X -{- d 



+ ab 
+ ac 
+ be 



X 4- ci^c 



3d 



x^ -{- a 


x^ + ab 


x^ -\- abc 


+ b 


+ ac 


+ abd 


+ c 


+ ad 


+ acd 


+ d 


+ bc 
+ bd 
+ cd 


+ bed 



X 4- c^crf 



These products, obtained by the common rule for algebraic 
multiplication, indicate the following laws : — 

1st. With respect to the exponents, we observe that the ex- 
ponent of X, in the first term, is equal to the number of bino- 
mial factors employed. In each of the following terms to the 
right, this exponent is diminished by 1 to the last term, where 
it is 0. 

2d. With respect to the co-efficients of the different powers 
of a:, that of the first term is 1 ; the co-efficient of the second 
term is equal to the sum of the second terms of the binomials ; 
the co-efficient of the third term is equal to the sum of the 
products of the different second terms, taken two and twoj 



ri 



i- 



CHAP. VII.] BINOMIAL THEOREM. 193 

the co-efficient of the fourth term is equal to the sum of their 
different products, taken three and three. 

Reasoning from analogy^ we might conclude that, in the pro- 
duct of any number of binomial factors, the co-efficient of the 
term which has n terms before it, is equal to the sum of the 
different products of the second terms of the binomials, t^ken 
n and n. The last term of the product is equal to the con- 
tinued product of the second terms of the binomials. 

In order to prove that this law of formation is general, sup- 
pose that it has been proved true for the product of m bino- 
mials. Let us see if it will com inn e to be true when the 
product is multiplied by a new binomial factor of the sam« 
form. 

For this purpose, suppose 
tf'-\-Ax'^-^-^Bx'^-'^-\-Cx'^-^ . . . +ifa;'"~«+i -{-Nx'^-"' -\- . . . + 11 
to be the product of m binomial factors ; iVi'"-" representing the 
term which has n terms before it, and Jifa;'"-»+^ the term which 
immediately precedes. 

Let X -\- Ic hQ the new binomial factor by which we multiply ; 
the product, when arranged according to the powers of ar,, 
will be 






x'^ -{- B 

■\- Ak 



-{-Bk 



r-OT — 2 



+ ... -\-N 
-\-Mk 



+ Uk',. 



from which we perceive that the law of the exponents is evi- 
dently the same. 

With respect to the co-efficients, we observe; 

1st. That the co-efficient of the first term is 1 ; and 

2d. That A-\- k^ or the co-efficient of a;'", is the sum of tJie 
second terms of the m -|- 1 binomials. 

3d. Since, by hypothesis, B is the sum of the different products 
of the second terms of the m binomials, taken two and two, and 
since A xk expresses the sum of the products of each of the 
second terms of the first m binomials by the new second term k ; 
therefore, B + Ak is the sum of the different products of ilie 
tecond terms of the m-\- \ binomials, taken two and two, 

13 



194 ELEMENTS OF ALGEBRA. [CHAP. VIT. 

In general, since JV expresses the sum of the products of the 
second terms of the m binomials, taken n and n, and M the sum 
of their products, taken n — 1 and n — I, therefore N + Mk, 
or the co-efficient of the term which has n terms before it, will be 
equal to the sum of the different products of the second terms 
of the m 4- 1 binomials, taken n and n. The last term is 
equal to the continued product of the second terms of the m -|- 1 
binomials. 

Hence, the law of composition, supposed true for a number m 
of binomial factors, is also true for a number denoted hjm-\- 1. 

But we have shown the law of composition for 4 factors , 
hence, the same law is true for 5 ; and being true for 5, it 
must be for 6, and so on; hence, it is general. 

136. Let us take the equation, 

{x -^ a) {x -\- b) {x -{- c) . ... =x"'-\- Ax"^^ + Bx"^"^ .... 

containing in the first member, m binomial factors. If we make 

a = 6 = c = c? . . . . &c., 

the first member becomes, 

{x + aY. 

In the second member the co-efficient of x^ will still be 1. 

The co-efficient of x^~^^ being a -^ b -{- c -{- d^ . . . will become 

a taken m times ; that is, ma. 

The co-efficient of a:"*-'^, being 

ab -\- ac -\- ad . . . . reduces to a^ -\- a^ -\- a^ . . . 

that is, it becomes a^ taken as many times as there are com 

binations of m letters, taken two and two, and hence reduce* 

(Art. 132), to 

m — 1 „ 

The oa-efficient of x"^^ reduces to the product of a^, multi- 
plied by the number of different combinations of m letters, 
taken three and three ; that is, to 

m — 1 m — 2 - . 



CHAP. VIL.] BINOMIAL THEOREM. 195 

Let us denote the general term, that is^ i/iie one which has 
n terms before it, by iVir'"'-". 

Tlien, the co-efficient iV will denote the sum of the products 
of the second terms, taken n and n ; and when all the 
second terms are supposed equal, it becomes equal to a" mul- 
tiplied by the number of combinations of m letters, taken 
n and n. Therefore, the co-efficient of the general term (Art. 

132), is 

F(m — n-{-l) 
i I L flrt . 

Qxn 
hence, we have, by making these substitutions, 

{x -f- a)^ = x^ -{■ max'^^ + m . — - — a^^"*-^ 

m — Im — 2^ _ F(m — n-{-l) 

^ 2 3 •••-r Q , n ...-r«, 

which is the binomial formula. 

The term 

F(m — n-h 1) 
Qn 
is called the general term^ because by making n = 2, 3, 4, &c., 
all the others can be deduced from it. The term which im 
mediately precedes it, is 

P P 

^n-l^m-n+l^ siuCC — 

H Q 

evidently expresses the number of combinations of m letters 
taken n —A and n — 1. Hence, we see, that 
p(^jn-n+ 1) 

Q X^ ' 

which is called the numeiical co-efficient of the general term, 

p 

is equal to the numerical co-efficient — of the preceding term, 

multiplied by m — ^ + 1, the exponent of x in that term, and 
divided by w, the number of terms preceding the required term. 

The simple law, demonstrated above, enables us to determine 
the numerical co-efficient of any term from that of the precedin^i; 
term, by means of the follow'ng 



196 ELEMENTS OF ALGEBRA. [CHAP. VIL 

RULE. 

The numerical co-efficient of any term after the firsts is forTued 
by multiplying that of the preceding term by the exponent of 
X in that term, and dividing the product by the number of 
terms which precede the required term. 

137. Let it be required to develop 

{x + ay. 

By applying the foregoing principles, we find, 
(x h ay = x^-\- Qax^ + l^a^x^ + 20(1^3 + iSa*^;^ + Qa^x + a». 
Having written the first term a;^, and the literal parts of the 
other terms, we find the numerical co-efficient of the second 
term by multiplying 1, the numerical co-efficient of the first 
term, by 6, the exponent of a; in that term, and dividing by 
1, the number of terms preceding the required term. To obtain 
the co-efficient of the third term, multiply 6 by 5 and divide 
the product by 2 ; we get 15 for the required number. The 
other numerical co-efficients may be found in the same manner 
In like manner, we find 

{x -f a)io z= x^^ + lOax^ -f ^oa^z^ + 120aV + 2\0a^x^ 
+ 252a5a;5 -f 2\0a^x^ -f 120a'a;3 + '^^ciH'^ + lOa^a: + a^^ 

138i The operation of finding the numerical co-efficients may 
be much simplified by the aid of the following principle. 

We have seen that the development of {x + a)"*, contains 
m-\-\ terms ; consequently, the term which has n terms after 
it, has m — n terms before it. Now, the numerical co-efficient 
of the term which has n terms before it is equal to the num- 
ber of combinations of m letters taken n in a set, and the 
numerical co-efficient of that term which has n terms after it, 
or m — n before it, is equal to the number of combinations of 
m letters taken m — n in a set ; but we have sho^vn (Art. 133) 
that these numbers are equal. Hence, 

In the development of any power of a binomial of the form 
[^ _j_ (x)* the numerical co-efficients of terms at equal distances from 
Hie two extremes^ are equal to each other. 



CHAP. VII.] BINOMIAL THEOREM. 197 

We see that this is the case in both of the examples above 
given. In finding the development of any power of a binomial, 
we need find but half, or one more than half, of the numerical 
co-efficients, since the remaining ones may be written directly 
from those already found. 

139. It frequently happens that the terms of the binomial, 
to which the formula is to be applied, contain co-efficlenta 
and exponents, as in the following example. 

Let it be required to raise the binomial 
Sa'^c — 2bd 
to the fourth power. 

Placing Sa^c = x and — - 2bd = y, we have 

{x -f yY = x^ + 4x^1/ + Qx^y^ + 4:xy^ + y* ; 
and substituting for x and y their values, we have 
(3a2c - 2hdY = {Sahy + 4 {Sa^y ( - 2bd) + 6 (3a2c)2 (- 2bdy 

+ 4 (3a2c) (- 2bdy + (- 2bdy, 
or, l^y performing the operations indicated, 

(3a2c - 2bdy = 81aV — 216a^c^d + 21Qa^c^^d^ — dQa'^cb^d^ 
-f 16b*d*, 

The terms of the development are alternately pjius and 
minus, as they should be, since the second term is — . 

140. A power of any polynomial may easily be found by 
means of the binomial formula, as in the following example. 

Let it be required to find the third power of 

a -{- b -{- c. 
First, put b -jr c =r d. 

Then ^ {a-{- b + cy = {a-{- dy = a^ + ZaH + 3ac?2 -|- d\ 
and by substituting for the value of c?, 

(g H- 5 -h c)3 = a3 + 3a26 + 3a62 + Ji 

3a2c -f 362c 4- 6aJc 
-f- 3ac2 -f- 36c2 



198 ELEMENTS OF ALGEBRA. ICHAP. VIL 

This developmei t is composed of the sum of the cubes of tht 
three terms, plus the sum of the results obtained by multiplying 
three times the square of each term, by each of the other terms in 
succession, plus six times the product of the three terms. 

To apply the preceding formula to the development of the 
cube of a trinomial, in which the terms are affected with co- 
efficients and exponents, designate each term by a single letter, 
and perform the operations indicated; then replace the letten 
introduced, by their values. 

From this rule, we find that 

(2a2 - 4a5 + 362)3 ^ ga^ _ 4Sa^b + U2a^b^ - 20Sa^P 
+ 198a26* - lOSa^s -t- 2756. 
The fourth, fifth, &;c., powers of any polynomial can be de- 
veloped in a similar manner. — ;U^ 

Extraction of the Cube Boot of Numbers, 

141 • The cube root of a number, is such a number as being 
taken three times as a factor, will produce the given number. 

A number whose cube root can be exactly found, is called a 
perfect cube ; all other numbers are imperfect cubes. 

The first ten numbers are, 
1, 2, 3, 4, 5, 6, 7, 8, 9, 10; 

and their cubes, 
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000. 

Conversely, the numbers in the first line are the cube roota 
of the corresponding numbers in the second. 

If we wish to find the cube root of any number less than 
1000, we look for the number in the second line, and if *t is 
there written, the corresponding number in the first line will be 
its cube root. If lb3 number is not there -wTitten, it will fall 
between two numbers in the second line, and its cube root 
will fall between the corresponding numbers in the first line. 
In this case the cube root cannot be expressed in exact parts 
of 1 ; hence, the given number must be an imperfect cube (Re- 
mark III, Art. 95). 



CHAP. VII.J C'JBE KOOT OF NUMBERS. 199 

If the given number is greater than 1000, its cube root will 
be greater than 10 ; that is, it will contain a certain number 
of tens and a certain number of units. 

Let us designate any number by iV, and denote its tens by 
a, and its units by b ; we shall have, 
iV^ m a + 6 ; whence, N^ = a^ + ^a?b -f 3a62 + b^ ; that is, 

The cube of a number is equal to the cube of the tens, plus three 
times the product of the square of the tens by the units, plus three 
times the product of the tens by the square of the units, plus the 
cube of the units. 
Thus (47)3= (40)3 4. 3 X (40)2 x 7 + 3 x40 X (7)2 -f (7)3 = 103823. 

Let us now reverse the operation, and find the cube root of 
103823. 



103 823 
64 



42 X 3 = 48 I 398'23 



47 
~8 



48 


47 


48 


47 


384 


329 


192 


188 


2304 


2209 


48 


47 


18432 


15463 


9216 


8836 



110592 103823 

Since the number is greater than 1000, its root will contain 
tens and units. We will first find the number of tens in the 
root. Now the cube of tens, giving at least thousands, we point 
off' three places of fig ires on the right, and the cube of the num- 
ber of tens will be i^und in the number 103, to the left of this 
pci lod. 

The cube root of the greatest cube contained in 103 being 4, 
this is the number of tens in the required root. Indeed, 103823 
is evidently comprised between (40)3 or 64,000, and (50)3 qj. 
125,000 ; hence, the required root is comprised between 4 tens 
and 5 tens : that is, it is composed of 4 tens, plus a certaiu 
number of un'ts less than ten. 



200 ELEMENTS OF ALGEBRA. ICHAP. VIL 

Having found the number of tens, subtract its cube, 64,- from 
103, and there remains 39, to which bring down the part 823| 
and we have 39823, which contains three times the product of 
the square of the tens by the units^ plus three times the product 
of the tens by the square of the units, plus the cube of the units. 

Now, as the square of tens gives at least hundreds, it follows 
that the product of three times the square of the tens by the 
.units, must be found in the part 398, to the left of 23, which 
is separated from it by a dash. Therefore, dividing 398 by 48, 
which is three times the square of the tens, the quotient 8 will 
be the units of the root, or something greater, since 398 is 
composed of three times the square of the tens by the units, and 
generally contains numbers coming from the two other parts. 

Vf e may ascertain whether the figure 8 is too great, by form- 
ing from the 4 tens and 8 units, the three parts which enter into 
39823 ; but it is much easier to cube 48, as has heeM done in 
the above table. Now, the cube of 48 is 110592, which is 
greater than 103823 ; therefore, 8 is too great. By cabjng 47, 
we obtain 103823 ; hence the proposed number is a }.erfeot cube, 
and 47 is its cube root. 

By a course of reasoning entirely analogous to that pvrsucd 
in treating of the extraction of the square root, we may s3igw 
that, when the given number is expressed by more than six 
figures, we must point off the number into periods of throe figui ^es 
each, commencing at the right. Hence, for the extraction of the 
cube root of numbers, we have the following 

RULE. 

I. Separate the given number into 2)eriods of three figures eacK, 
beginning at the right hand ; the left hand period will often con 
tain less than three places of figures. 

II. Seek the greatest perfect cube in the first 2^criod, on the left, 
and set its root on the right, after the manner of a quotient iv 
division. Subtract the cube of this number from the first period^ 
and to the remainder bring down the first figure of the next period^ 
and call this number the dividend. 



(*HAP. VII.J EXTRACTION OF ROOTS. 201 

III. Take three times the square of the root just found for a 
divisor, and see how often it is contained in the dividend, and 
place the quotient for a second figure of the root. Then cube the 
number thus found, and if its cube be greater than the first two 
periods of the given number^ diminish the last figure by 1 ; but 
if it be less, subtract it from the first two periods, and to the 
remainder bring down the first figure of the next period, for a new 
dividend. 

IV. Take three times the square of the whole rcot for a new 
divisor, and seek how often it is contained in the new dividend ; 
the quotient will be the third figure of the root. Cube the number 
thus found, and subtract the result from the first three periods 
of the given number, and proceed in a similar way for all the 
periods. 

If there is no remainder, the number is a perfect cube, and the 
root is exact : if there is a remainder, the number is an imper- 
fect cube, and the root is exact to within less than 1. 

EXAMPLES. 

1. 3/48228544 Ans. 3G4. 

2. y 27054036008 Jns. 3002. 

3. ^483249 Ans. 78, with a remainder 8697. 

4. 3/91632508641 Ans. 4508, with a remainder 20644129. 

5. y 32977340218432 Ans. 32068. 

Extraction of the N^^ Boot of N'umhers. 

142. The n*^ root of a number is such a number as being 
taken n times as a factor will produce the given number, n being 
in\ positive whole number. When such a root can be exactly 
found, the given number is a perfect n*^ power ; all other num- 
bers are imperfect n^^ powers. 

Let iV" denote any number whatever. If it is expressed by 
less than /i + 1 figures, and is a perfect n*'^ power, its qi^^ root ' 
will be expressed by a single figure, and may be found by 



202 ELEMENTS OF ALGEBRA. [CHAP. y£L 

means cf a tabi3 containing the n*^ powers of the first ten 
numbers. 

If the number is not a perfect ti*^ power, it will fall between 
two n*^ powers in the table, and its root will fall between the 
n*^ roots of these powers. 

If the given number is expressed by more than n figures, 
its root will consist of a certain number of tens and a certain 
number of units. If we designate the tens of the root by a, 
and the units by b, we shall have, by the binomial formula, 

^ \ 

iV^r= (a + 6)« = a« + na'^-iJ -f- n — -— a"-262 _f-^ &;c. ; 

that is, the proposed number is equal to the n^^ power of the 
tens, plus n times the product of the n — V^ power of the tens 
by the units, plus other parts which it is not necessary to 
consider. 

Now, as the n*^ power of the tens, cannot be less than 
1 followed by n ciphers, the last n figures on the right, cannot 
make a part of it. They must then be pointed off, and the n** 
root of the greatest n*^ power in the number on the left will 
be the number of tens of the required root. 

Subtract the n*^ power of the number of tens from the num 
ber on the left, and to the remainder bring do^^^l one figure of 
the next period on the right. If we consider the num.ber thus 
fuund as a dividend, and take n times the {n — 1)^^ power 
of the number of tens, as a divisor, the quotient -will evidently 
be the number of units, or a greater number. 

If the part on the left should contain more than n figures, the 
n figures on the right of it, must be separated from the rest, 
and the root of the greatest n*^ pow-er contained in the part 
on the left extracted, and so on. Hence the following 

EULE. 

I. Separate the number J^ into periods of n figures each, he 
ginning at the right hind; extract the n*^ root of the greatest 
perfect n*^ power contained in the hft hand period, it will be th$ 
first figure of the root. 



CHAP. VII.] EXTRACTION OF ROOTS. 20.^ 

II. Subtract this n*^ power from the left hand period and bring 
down to the right of the remainder the first figure of the next 
period, and call this the dividend. 

in. Form the n — 1 power of the first fig we of the root, mul- 
tiply it by n, and see how often the product is contained in the 
dividend: the quotient will be the second figure of the root, or 
something greater. 

IV. Raise the number thus formed to the n*^ power, then sub- 
tract this result from the tiuo left-hand periods, and to the new 
remainder bring down the first figure of the next period : then 
divide the number thus formed by n times the n — 1 power . of 
the two figures of the root already found, and continue this opera- 
tion until all the periods are brought down. 

EXAMPLES. 

1. What is the fourth root of 531441 1 

53 1441 I 27 
2*= 16_ 

4 X 23 =. 32 1 371 
(27)* = 531441. 

We first point off, from the right hand, the period of four 
figures, and then find the greatest fourth root contained in 53, 
the first period to the left, which is 2. We next subtract the 
4th power of 2, which is 16, from 53, and to the remainder 
37 we bring down the first figure of the next period. We 
then divide 371 by 4 times the cube of 2, which gives 11 for 
a quotient : but this we know is too large. By trying the num- 
bers 9 and 8, we find them also too large : then trying 7, we 
find the exact root to be 27. 

143. When the index of the root to be extracted is a multiple 
of two or more numbers, as 4, 6, . . . &:c., the root can be ob- 
tained by extracting roots of more simple degrees, successively. To 
explain this, we will remark that, 

{a^Y = 0^ X a3 X c'3 x «3 = ^3 + 3 + 3 + 3 _ a^y^^ _ ^^2^ 

and, in general, from the definition of an exponent 
[a^Y — a^ X a^ X a"^ X a"^ . . . = a'^X": 



204 ELEMENTS OF ALGEBKA. [CHAP. VIL 

hence, the n^^ power of the m*^ power cf a number is equal to tki 
mn*^ power of this number. 

Let us see if the converse of this is alst true. 



Let V V^ "" ^ ' 



then raising both members to the n*^ power, we have, from the 
definition of the n^^ root, 

and by raising both members of the last equation to the m^^ power 

a — b""^. 
Extracting the mn*'^ root of both members of the Igst equation, 

1 "" fa = b • 

we have, V ' 



and hence, \/'y'~a = '"'i/a', 

since each is equal to b. Therefore, the n*^ root of the rnJ^ rcoi 
of any number^ is equal to the mn^^ root of that number. And 
in a similar manner, it might be proved that 

By this method we find that 



I. 1/256 = \/ V256= /16 



2. 6^2985984 = Wy/ 2985984 = ^1728 = 12. 



3. 



^1771561 =: x/y 1771561 =: IL 



4. y 1679616 = 1^1296 =\/ ^ 1296 = 6. 

Remark. — Although the successive roots may be extracted in 
%ny order whatever, it is better to extract the roots of the lowest 
degree first, for then the extraction of the roots of the higher 
degrees, which is a more complicated operation, is effected upon 
numbers containing fewer figures than the proposed number. 



CHAP.. VI I.] EXTRACTION OF ROOTS. 205 

Extraction of Roots hy Approximation. 

144. When it is required to extract the n*^ root of a number 
which is not & perfect n^^ power ^ the method already explained, will 
give only the entire part of the root, or the root to within less 
than 1. As to the part which is to be added, in order to com 
plete the root, it cannot be obtained exactly, but we can approx- 
imate to it as near as we please. 

Let it be required to extract the n*^ root of a whole number, 

denoted by a, to within less than a fraction — ; that is, so near, 

♦nat the error shall be less than — . 

p 

We observe, that we can write 

op" 

a =. . 

p- 

If we denote by r the root of the greatest perfect n** power in 
ap*^ the number ~ = a, will be comprehended between — and 

(y _j_ \\n 

~ ^j-^ ; therefore, the y a will be comprised between the 

r r + 1 
two numbers — and ; and consequently, their difference 

1 . r 

— will be greater than the difference between — and the true- 

P P 

r 
root. Hence, — is the required root to within less than the 

p 

fraction — : hence, 
P 

To extract the n^^ root of a whole number to within less than 
a fraction — , multiply the number by p^ ; extract the n*^ root of 
(he product to within less than 1, and divide the result by p. 

Extraction of the n*^ Root of Fractions, 

145. Since the n*^ power of a fraction is formed by raising 
both terms of the fraction to the n*^ power, we can evidently 
find the n*^ root of a fraction by extracting the n*^ root of 
both terms. 



Lot — - 



206 ELEMENTS OF ALGEBRA. [CHAP. VII. 

If both terms are not perfect n*'* powers, the exact n*^ root 
cannot be found, but we may find its approximate root \j9 
within less than the fractional unit, as follows: — 
a 
— represent the given fraction. If we multiply botb 

terms by 

5"-^, it becomes, -— = , 

6 6" 

Let r denote the •«** root of the greatest ti** power in at*"**, 

then -— — will be comprised between ■— and ^^ — ; 

T a 

and consequently, — will be the n^^ root of — - to within les» 



than the fraction — ; therefore, 

Multiply the numerator by the {n — iy^ power of the denomi 
nator and extract the n*^ root of the product: Divide this root 
by the denominator of the given fraction, and the quotient will 
he the approximate root. 

When a greater degree of exactness is required than that 
indicated by — , extract the n^^ root of aJ"-^ to withir Any 

1 r' r' 

fraction — : and designate this root by — . Now, sinci — 
p' ^ ^ P ' P 

is the root of the numerator to within less than — , it fol » ws, 

p 

r' . 1 

that -— is the true root of the fraction to within less thaL -— 
hp op 

EXAMPLES. 

1. Suppose it were required to extract the cube root (^15 

to within less than — . We have 

15 X 123 3= 15 X 1728 = 25920. 
Now, the cube root of 25920, to within less than I is £9 
hence, the required root is, 

13~" 12' 



I 



CHAP. VII.] EXTRACTION OF ROOTS. 207 

2. Extract the cube root ©f 47, to within less than — . 

We have, 

47 X 203 = 47 X 8000 = 376000. 

Now, the cube root of 376000, to within less than 1, is 72 ; 

, tysy 12 1 

hence, 3/47 zr: — = 3 —-, to within less than — . 

3. Find the value of ?/25, to within less than .001. 

To do this, multiply 25 by the cube of 1000, or 1000000000, 
which gives 25000000000. Now, the cube root of this number, 
is 2920 ; hence, 

^/25 = 2.920 to within less than .001. 
Hence, to extract the cube root of 9 whole number to 
within less than a given decimal fraction, we have the following 

RULE. 

Annex three times as many ciphers to the number^ as there are 

decimal places in the required root ; extract the cube root of the 

number thus formed to within less than 1, and point off from 

the right of this root the required number of decimal places. 

146i We will now explain the method of extracting the cube 
root of a decimal fraction. 

Suppose it is required to extract the cube root of 3.1415. 

Since the denominator, 10000, of this fraction, is not a per 
feet cube, make it one, by multiplying it by 100 ; this is equiva 
lent to annexing two ciphers to the proposed decimal^ which then 
becomes, 3.141500. Extract the cube root of 3141500, that is, 
of the number considered independent of the decimal point to 
within less than 1 ; this gives 146. Then dividing by 100, o) 
^ 1000000, and we find, 

3/3.1415 - 1.46 to within less than 0.01. 
Hence, to extract the cube root of a decimal fraction, we haTi 
the following 



208 ELEMENTS OF ALGEBRA. I CHAP. VIL 

RULE. • 

A7inex ciphers till the whole number of decimal places is equal 
to three times the number of required decimal places in the root. 
Then extract the root as in whole numbers, and point off the re- 
quired number of decimal places. 

To extract the cube root of a vulgar fraction to -within less 
than a given decimal fraction, the most simple method is, 

To reduce the proposed fraction to a decimal fraction, continuing 
the division until the number of decimal places is equal to three 
times the number required in the root. 

The question is then reduced to extracting the cube root of 
a decimal fraction. 

Suppose it is required to find the sixth root of 23, to 
within less than 0.01. 

Applying the rule of Art. 144 to this example, we multiply 
23 by (100)^, or amiex twelve ciphers to 23 ; then extract the 
sixth root of the number thus formed to within less than 1, 
an<? di\nde this root by 100, or point off two decimal places 
on the right : we thus find, 

23 = 1.68, to within less than 0.01. 



EXAMPLES. 

1. Find the ^473 to within less than -}-q. Ans. 7 J. 

2. Find the ^/79 to within less than .0001. Ans. 4.2908. 
3; Fmd the « /13 to within less than .01. Ans. 1.53. 



4. Find the ^3.00415 to \\dthin less than .0001. 

Ans. 1.4429. 

5. Find the ^0.00101 to within less than .01. 

Ans. 0.10. 

6. Find the ^/JT to within less than .001. Ans. 0.824. 



CUAP. VII.J EXTRACTION OF ROOTS. 209 

Extraction of Roots of Algebraic Quautiiies, 

147t Let us first consider the case of monomials, and in order 
to deduce a rule for extracting the n^^ root, let us examine the 
law for the formation of the n*^ power. 

From the definition of a power, it follows that each factor 
of the root will enter the power, as many times as there are 
units in the exponent of the power. Tliat is, to form the w** 
power of a monomial, 

We form the w'* power of the co-efficient for a new co-efficient^ 
and write after this, each letter affiuUd wolh an exponent equal to 
ft times its primitive exponent. 

Conversely, we have for the extraction of the w** root of a 
monomial, the following 

RULE. 

Extract the n'* root of the numerical co-efficient for a new co- 
efficient, and after this write each letter affected with an exponent 

equal to — th of its exponent in the given monomial; the result 
will he the required root. 



Thus, ^/ei^Wi^ = 4a35c2; and ^^ l%a%^'^c^ = "XaWc. 

From this rule we perceive, that in order that a monomial 
may be a perfect n'* power: 

1st. Its co-efficient must be a perfect w'* power; and 
2d. The exponent of each letter must be divisible by n. 
It will be shown, hereafter, how the expression for the rout 
of a quantity, which is not a perfect power, is reduced to its- 
simplest form. 

148. Hitherto, in finding the power of a monomial, we have 
paid no attention to the sign with which the monomial may be 
affected. It has already been shown, that whatever be the sigu 
of a monomial, its square is always positive. 

14 



210 ELEMENTS OF ALGEBRA. LCHAP. TIU 

Let n be any whole number; then every power of an even 
degree, as 2/1, can be considered as the n*^ power of the square ; 
that is, {a?)* = ^^" • hence, it follows, 

That every power of an even degree, will be essentially posi 
/ii'«, whether the quantity itself be positive .>r negative. 

Thus, (±2a263c)* = 4- Ua%^h\ 

Again, as every power of an uneven degree, 2n -\- I, is but 
Llie product of the power of an even degree, 2n, by the first 
power ; it follows that. 

Every power of a monomial, of an uneven degree, has the same 
sign as the monomial itself. 

Hence, {+ ^a^f = -\- Q^%^ -, and {^ - ^^f - — Ua%\ 

From the preceding reasoning, we conclude, 

1st. That when the index of the root of a monomial is uneven, 
the root will be affected with the same sign as the monomial. 

Thus, 



3/+8a3 = + 2a ; ij - ^a^ = - 2a ; y - S2a^^b^ = - 2a^K 

2d. When the index of the root is even, and the monomial a 
positive quantity, the root has both the signs -\- and — . 

Thus, t^81a*6i2^ ± 3^53 . ^ q^^is ^ _h 2a\ 

3d. When the index of the root is even, and the monomial vega,' 
live, the root is impossible; 

For, there is no quantity which, being raised to a power of 

au even degree, will give a negative result. Therefore, 

tr^a. \n^. \r^c. 



are symbols of operations which it is impossible to execute 
Tliey are imaginary expressions. 

EXAMPLES. 

1. What is the cube root of %a%H^'^% Ans, 2a%C^, 

2. What is the 4th root of 81a*68ci«? Ans. Sab^c*. 
a. What is the 5th root of — ^2a^c^°d^^ ? Ans. — ^ac^d^. 
4. What is the cube root of — 'i2ba^^c^l Ans. — baWc. 



CHAP. VII.] EXTEACTION OF ROOTS. 211 

Extraction of the w'* Root of Polynomials. 

148. Let N denote any polynomial whatever, arranged with 
reference to a certain letter. Now, the ^i'* power of a poly- 
nomial is the continued product arising from taking the poly, 
nomial n times as a factor: hence, the first term of the pro- 
duct, wht^n arranged with reference to a certain letter, is th ' 
n** power of the first term of the polynomial, arranged with 
reference to the same letter. 

Therefore, the n** root of the first term of such a product, 
will be the first term of the n** root of the product. 

Let us denote the first term of the »** root of N by r, 
and the following terms, arranged with reference to the lead- 
ing letter of the polynomial, by r', r'\ r"\ &c. We shall 
have, 

N= (r + r' -\-r" -\- , . &c.)» ; 

or, if we designate the sum of all the terms after the first 
Dv «, 

iV= (r + s)" = r" + nr^-'^s -\- &c., 

= r« + wr»-i(r' + r" + &c. ) -f (fee. 

Jf now, we subtract r" from iV, and designate the remainder 
by i?, we shall have, 

i2 = iV — r" = wr"- V + nr«- V" + &c., 

which remainder will evidently be arranged with reference to 
the leading letter of the polynomial ; therefore, the first term 
will contain a higher po\\'er of that letter than either of the 
succeeding terms, and cannot be reduced with any of them. 
Hence, if we divide the first term of the first remainder, by 
n times the {n — 1)'* power of the first term of the root, the 
quotient will be the second term of the root. 

If now, we place r -f- r' = w, and denote the sum of the suo. 
feeding terms of the root by s\ we shall have, 

JV= (w -I- s'Y = w 4- nu^'-^s' + &c 



212 ELEMENTS OF ALGEBRA,- [CHAP. VII. 

If now, we subtract w" from iV^, and den:5te the remainder bj^ 
22'. we shall have, 

B' =z N— w" = n{r + r')"-V + &c., 

_ 7?yn-l(// _|_ r>" ^ &;C. ) + &C., 

K we divide the first term of this remainder by n times 
the {n — 1)*'' power of the first term of the root, we shall 
have the third term of the root. If we continue the operation, 
we shall find that the first term of any new remainder, divided 
by n times the (n — 1)'* power of the first term of the root, 
will give a new term of the root. 

It mrty be remarked, that since the first term of the first 
remainder is the same as the second term of the given poly- 
nomial, we can find the second term of the root, by dividing 
the setond term of the given polynomial by n times the 
{ih — 1)'* power of the first term. 

ITent^'e, for the extraction of the n'* root of a polynomial, 
we have the following 

RULE. 

I. Arrange the given polynomial with reference to one of its letters^ 
mid extract the n'* root of the first term; this will he the first 
term of the root. 

II. Divide the second term hy n times the (n — 1)'* power of the 
first term of the root ; the quotient will he the second tern>. of the root. 

III. Subtract the w'* power of the sum of the two terms already 
found from the given polynomial, and divide the first term of 
the remainder hy n times the {n — 1)** power of the first term of 
the root; the quotient will he the third term of the root. 

IV. Continue this operation till a remainder is found equal to 
0, o^, till one is found whose first term is not divisible hy n times 
tlie {n — \y^ power of the first term of the root: in the former case 
the root is exact, and the given polynomial a perfect w'^ power ; 
in the latter case, the polynomial is an imperfect n*^ power. 



CHAP. VII.] EXTEACTION OF ROOTS. 213 

149. Let us apply the foregoing rule to the following 

EXAMPLES. 

1. Extract the cube root of x^-(jx^-i-\5x*—20x^-i-16x^-Qx-\-\. 

xG-Qx^-{.l5x*—20x^-hl5x'^-Qx+ l \x^-2x-^l 

(x^-2x Y=x^-(jx^+12x*— Sx^ Sx* 

1st rem. 3x*—12x^+ &;c. 

(x^-2x-i-iy=x^-Gx^-\-16x*—20x^+l5x^-Gx+l, 
In this example, we first extract the cube root of x^, which 
gives a;2, for the first term of the root. Squaring x^, and mul- 
tiplying by 3, we obtain the divisor 3a;* : this is contained in 
the second term — Qx^, —2x times. Then cubing the part of 
the root found, and subtracting, we find that the first term of 
the remainder 3^*, contains the divisor once. Cubing the whole 
root found, we find the cube equal to the given polynomial. 
Hence, a;^ — 2a; + 1? is the exact cube root. 

2. Find the cube root of 

a;6 + 6^5 _ 40a;3 _|. 96a; — 64. 

3. Find the cube root of 

8a;6 - 12a;5 -f 30a;* - 25a;3 + 30a:2 _ i^x ^ 8. 

4. Find the 4th root of 16a*- ma?x + 2\Qa?x'^ — 2\Qax^ + 81a» 

16a*-96a3a;+216a2a;2-216aa;3+81a;4 2a-3a; 
(2a-3a;)*= 16a*-96a%+216a2a;2-216aa;34-81a;* 4x(2a)3=32a3. 

We first extract the 4th root of 16a*, which is 2a. We then 
raise 2a to the third power, and multiply by 4, the index of the 
root ; this gives the divisor o2a?. This divisor is contained in 
the second term — QGa^a;, — 3a; times, which is the second term 
cf the root. Raising the whole root found to the 4th power 
we find the power equal to the given polynomial. 

5. What is the 4th root of the polynomial, 

81a«c* 4- 165*c?* — ^QahbH^ -^ 2\(ja^c%d -f- 216a*c2^2£;2. 

6. Find the 5th root of 

32a;5 — 80a;4 4. gOa;' - 40a;2 -f 10a; - 1. 



214 



ELEMENTS OF ALGEBRA. 



LCHAP. vn. 



Transformation of Radicals of any Degree, 

150t The principles demonstrated in Art. 104, are general 
For, let IL/a" and 'l/6J be any tvro radicals of the n'* degree, 
and denote their product ^j p. We shall have, 

V^xy^=i) - - - (1). 

By raising both members of this equation to the n** power, 
we find 

{\/~^Y X (^A)" =i?", or ab =^" ; 
whence, by extracting the n'* root of both members, 

■\/ab=p . . . (2). 

Since the second members of equations (1) and (2) are the 
same, their first members are equal, whence, 

^J~a X \r^— ^J~oih : hence, 
1st. The product of the n*^ roots of two quantities^ is equal to 
the n*^ root of the product of the quantities. 

Denote the quotient of the given radicals by q^ we shall have 



«/6 

and by raising both members to the n'* power, 



- (1); 



or 



= ?": 



whence, by extracting the n'* root of the two members, we 
have, 

n / n 

(3). 



- = y . . . 

The second members of equations (1) and (2) being the same, 
their first members are equal, giving 

'L/a * Fa' 

- — = Y — ; hence, 

2d. The quotient of the n** roots of two quantities^ is equal to 
titc w'* root of the quotient of the quantities. 



CHAP. VII.l TRANSFORMATION OF RADICALS. 215 

151, Let us apply the first principle of article 150, to the 
simplification of the radicals in the following 

EXAMPLES. 

1. Take the radical 3^54a'*6V. This may be written, 

\J 54a*63c2 = yriaW- X l/2ac'^ = Sab \J^mc'^, 

2. In like manner, 

^"802"== 2 y^; and ^^ ^%a^h\^ = 'Xah'^c \fs^^ ; 

3. Also, 

yi92a^6ci2 ^ ^64a''ci2 x \fSd) = 2ac^^^/3^. 

In the expressions, Sab\/2ac'^, 2\/~a^, 2ab'^c */ Sac^, 

each quantity placed before the radical, is called a co-efficieiU 
of the radical. 

Since we may simplify any radical in a similar manner, we 
have, for the simplification of a radical of the 71'* degree, the 
follo^ving 

RULE. 

Resolve the quantity under the radical sign into two factors^ one 
of which shall be the greatest perfect n^^ power which enters it; 
extract the n*^ root of this factor, and write the root without t?ie 
radical sign, under u'hich, leave the other factor. 

Conversely, a co-efficient may be introduced under the radical 
airn, by simply raising it to the w.'* power, and writing it as a 
factor under the radical sign. 

Thus, Sab \f%I^ = \j21a'^b^ X ^^2^ = y^54a*6V. 

152. By the aid of the principles demonstrated in article 143, 
we arc enabled to make another kind of simplification. 

Take, for example, the radical ^4a2; from the principles re« 



ferred to, we have, 



4a2 



-\//^^ 



216 ELEMENTS OF ALGEBRA. [CHAP. VII. 

and as the quantity under the radical sign of the second degree 
is a perfect square, its root can be extracted : hence, 

In like manner, 



In general, 

/ — ^ / ^ I — 

that is, -sN-hen the index of a radical is a multiple of any number 
n, and the quantity under the radical sign is an exact 7^'* power, 
We can, without changing the value of the radical, divide its index 
by ,n, and extract the ti'* root of the quantify under the sign, 

153. Conversely, The index of a radical may be multiplied by 
any number, provided we raise the quantity under the sign to a 
power of which this numher is the exponent. 

For, since a is the same thing as 1,/a", we have, 

154i The last principles enable us to reduce two or more 
radicals of different degrees, to equivalent radicals having a com- 
mon index. 

For example, let it be required to reduce the two radicals 

Ya and \f{a -\-b) 



3 

to the same index. 

By multiplying the index of the first by 4, the index of the 
second, and raising the quantity 2a to the fourth power; then 
multiplying the index of the second by 3, the index of the 
first, and cubing a -\- b, the value of neither radical will b<» 
changed, and the expressions will become 

y^ = iy^2^ = ly^ie^; and y (a + 6) = ^y (a + 6)3, 

and similarly for other radicals : hence, to reduce radicals to a 
common index, we have the following 



CHAP. VII.] TRANSFORMATION OF RADICALS. 217 

RULE. 

Multiply the index of each radical hy the product of the indices 
of all the other radicals^ and raise the quantity under each radical 
sign to a power denoted hy this product. 

This rule, which is analogous to that givec for the reduction 
of fractions to a common denominator, is susceptible of similar 
modifications. 

For example, reduce the radicals 

to a common index. 

Since 24 is the least common multiple of the indices, 4, 6, and 
8, it is only necessary to multiply the first by 6, the second by 
4, and the third by 3, and to raise the quantities under each rad 
ical sign to the 6th, 4th, and od powers, respectively, which gives 

Addition and /Subtraction of Radicals of any Degree. 

155, We first reduce the radicals to their simplest form by 
the aid of the preceding rules, and then if they are similar^ in 
order to add them together, we add their co-efficients^ and after 
Utis sum write the common radical; if they are not similar, th« 
addition can only be indicated. 

Thus, 3^+23^r=5^. 

EXAMPLES. 

1. rind the sum of ^ 4Bab'^ and b^lba. Ans. QSVSa. 

2. Find the sum of S^^^J^ and 2^^. Ans. 5^/2a. 

3. Find the sum of 2.^/45 and o ^. Ans. 9^5. 

155^'. In order to subtract one- radical from another when 
tliey are similar, 

Subtract the co-efficient of the subtrahend from the co-efficient of 
Hkb mi'^.uend^ and write V is difference before the common radical^ 



■/r 



218 ELEMENTS OF ALGEBRA. 'CHAP. VIL 

Thus, 3a yT - 2c \fh = (3a - 2c) y^; 

but, 2ab yfcd — ^ab J~c are irred jcible. 

1. From y8a36+ IGa* subtract \fW^\^2d^. 

Ans, (2a - h) ^ 6 + 2a. 

2. From S^.^/4a^ subtract 2^/2a. ^/is. ^/2a. 

Multiplication of Radicals of any Degree. 

156. We have shown that all radicals may be reduced to 
equivalent ones having a common index; we therefore suppose 
this transformation made. 

Now, let a^J~b and C^Td denote any two radicals of the 
same degree. Their product may be denoted thus, 

a^yix cyT; 
or since the order of the factors may be changed without affect- 
ing the value of the product, we may write it, 

m X "y/^X ^fd or (Art. 150), since "U d x y^= \/^; 
we have finally, 

a?/6"x cird = ac\rbd\ 
hence, for the multiplication of radicals of any degree, we have 
the fol]o^ving 

RULE. 

I. Reduce the radicals to equivalent ones having a common index. 

n Multiply the co-efficients together for a new co-efficient; after 
this write the radical sign with the common index, placing under 
it the product of the quantities under the radical signs in the two 
factors; the result is the product required. 

EXAMPLES. 

1. The product 

_ 6a'(a' + &^) 



CHAP. VII.J TRANSFORMATION OF RADICALS. 219 

2. Tlie product 

3at/8a2~x 2b l^/i^ = Gab ^ 32a*c = \2o?h ^^/2c. 

3. The product 

3 3 /T 1 3 /T _ 3 3 /T 

Tvy ^TVT-i6V2r 

4. The product 

3a 5/6" X 56 \f2c = 15c.^ X ^y^Si*^. 
4 /r 3 Af 

Am. ^^. 



5, Multiply ^2 X \fZ by \ / — X 



6. Multiply 2^15 by 3?/l0. 



^715. Gy" 337500, 
7. Multiply 4yy by 2sJ ^, 

10 /27 



^...8 ,,^^^. 



8. Multiply y^ y^ and y^, together. 

^/i5. ly 648000. 

' /4~ 3 /y 

9. Multiply W-o-j V "o" ^^^ ^V ^' *^a®*^^^- 

10. Multiply (4/l+5y/T) by (/I+2^1). 



43 . 13 
3 



Am, ^+—^42. 



Division of Radicals of any Degree, 

157« We will suppose, as in the last article, that the radicals 
have been reduced ^o equivalent ones having a common index. 

Let cL'lfb^ and Clfd represent any two radicals of the 
r* * degree. The quotient of the first by the second may be 
written, 



220 ELEMENTS OF ALGEBRA. [CHAP. VIL 

v^ • rv 

or, since X_. = * /_. (Art. 150), re have, 

\fd ^ ^ 

Hence, to divide one radical by another, we have the fol 
lowing 

RULE. 

I. Reduce the radicals to equivalent ones having a common ijides^. 

II. Divide the co-efficient of the dividend hy thai of the divi- 
so^r for a new co- efficient ; after this write the radical sign with 
the common index, and place under it the quotient obtained hy 
dividing the quantity under the radical sign in the dividend hy that 
in the divisor ; the result will he the quotient required. 

EXAMPLES. 



1 . What is the quotient of c l/aW- ■\- 6* divided by d \ /- 

c^ IJaW + 6^ ^ g 3 /S6 (^2^2 ^ 54) ^ 2ch 3 /g^ + h^ 
d ^ 3 /a2 _ 52 ~T V a2 — 62 -~d\/a^ — h'^' 



3 /a2-h2^ 



86 
2. Divide 2y^x3y4 by ^ ^/^X ^. 



A?is. 4 12/ 288. 



3, Divide y(/1 X 2 3^3 by ^4 3^2 X ysT 



4. Divide 1/J by (72 + 3^). 

5. Divide 1 by \/~^ + \/T. 



^"^- y'l/f- 



^n.. 1. 



tfd'~\fd^-\-^Ja^-^J~^ 

Ans. y. 11 . V V 

a-b 



CHA.P Vir.I TRANSFORMATION OF RADICALS 221 

6. Divide %fa> -\- \/T by \f^—\fb. 

a + i + 2ya6 4- 2 1^63^ + 2 */a^ 

Ans. Y- — 

a — b 

Formation of Powers of Radicals of any Degree, 

158» Let a 'iTb represent any radical of the n^^ degree, 
rhen we may raise this radical to the m^^ power, by taking 
it m times as a factor; thus, 

a'^fhXa'tJ'b a \/^ 

But, by the rule for multiplication, this continued product is 
equal to a^ ^T^'y whence, 

(a y^)*" = a*" y^ - - - - (1). 
We have then, to raise a radical to any power, the following 

RULE. 

liaise the co-efficient to the required power for a new co-efficient ; 
after this write the radical sign with its primitive index, placing 
under it the required power of the quantity under the radical 
sign in the gipen expression ; the result will be the power required, 

EXAMPLES. 

1. {\f^f = 1y/J^y = t/TO^ = 2at/^ = 2a y^. 

2. (33/2^)5 = 35 3/(2^)5 = 243 3/32^ =48Ga3ylL^. 

When the index of the radical is a multiple of the expo- 
nent of the power to which it is to be raised, the result can 
be simplified. 

For, i/^2a"==\yy^a" (Art. 152): hence, in order to square 

*/ 2a, we have only to omit the first radical sign, which gives 

{\/2^y =./2^. 

Again, to square i/^, we have i/36 = \/i/^* ^^^^e, 
(^/^)2 = y^; hence, 



222 ELEMENTS OF ALaEBRA, fCHAP. VIL 

When the index of the radical is divisible by the exponent of 
the power to which it is to be raised, perform the division, leaving 
the quantity under the radical sign unchanged. 

i^ Extraction of Roots of Radicals of any Degree, 

159. By extracting the m^^ root of both members of equa- 
tion (1), of the preceding article, we find, 



Va'" X \/b^'=zayF', 
Whence we see, that to extract any root of a radical of any 
degree, we have the following 

RULE. 

Extract the required root of the co-efficient for a -new co-efficient ; 

after this write the radical sign with its primitive index, under 

which place the required root of the quantity under the radical 

sign in the given expression; the result will be the root required. 

EXAMPLES. 

1. Find the cube root of 81S^/27. Ans. 2t/3. 

2. Find the fourth root of t^\/256. Ans. -^ifi. 

159*. If, however, the required root of the quantity under the 
radical sign cannot be exactly found, we may proceed in the 
following manner. If it be required to find the m'* root of 
c^Td^ the operation may be indicated thus, 

yc r^d = y7 y '/of ; 

m I ~Z. — 

but yj tJ d = '"'l/ c?, whence, by substituting in the previous 
equation, 

Consequently, when we cannot extract the required root of the 
quantity under the radical sign. 



CHAP. VII.] TRANSFORMATION OF RADICALS. 223 

Extra<t the required rqot of the co-efficient for i new co-efficient ', 
after this^ write the radical sign, with an index equal to the pro- 
duct of its primitive index by the index of the required rooty 
leaving the quantity under the radical sign unchanged, 

EXAMPLES. 



1. yV^c^^V^; and, sj\/hc^'^c. 

When the quantity under the radical is a perfect power, of 
the degree of either of the roots to be extracted, the result can be 
simplified. 

Thus, 1/ t/S^ = y \f^ = \/Ya. 



In like manner, J y'O^ = ^ ^ Qa^ = y^. 

2. Find the cube root of — J~Z. Ans. — l/3. 

3. Fmd the cube root of — y^2^. Ans. — ^^2^. 

Different Roots of the same Power. 

160« ITie rules just demonstrated depend upon the principle, 
that if two quantities are equal, the like roots of those quantities 
are also equal. 

This principle is true so long as we regard the term root 
in its general sense, but when the term is used in a restricted 
sense, it requires some modification. This modification is parti- 
cularly necessary in operating upon imaginary expressions, which 
are not roots, strictly speaking, but mere indications of opera- 
tions which it is impossible to perform. ' Before pointing out 
these modifications, it will be shown, that every quantity has 
more than one cube root, fourth root, &ic. 

It has already been shown, that ^WQ,ry r[uantity has two square 
roots, equal, with contrary signs. 



224 ELEMENTS OF ALGEBRA. [CHAP. VIL 

1. Let X denote the general expression for the cube root of 

a, and let "p denote the numerical value of this rootj we have 

the equations 

x^ = a, and x^ = p^. 

The last equation is satisfied by making x =2). 
Observing that the equation x^=p^ can be put under the form 
e^ — j)^ =z 0, and that the expression x^—p^ is divisible by 
T — JO, giving the quotient, x'^ -\- px -{- p'^^ the above equation can 
be placed under the form 

(a; — p) (a;2 4-j9a; +^2^ = 0. 
Now, every value of x that will satisfy this equation, will 
satisfy the first equation. But this equation can be satisfied by 
supposing 

ar— ^ = 0, whence, x-=.p\ % 

or by supposing 

x^ -\- px -^^ p"^ =1 0, 
from which we have. 



P_^L / — ^ ^^ /-i±V~3) 

2 ~ 2 



--±-./-3, or x=p{ ^ j; 



hence, we see, that there are three different algebraic expressions 
for the cube root of a^ viz: 

^' A — ^ — > ""^ A — 2 — } 

2. Again, solve the equation 

x^=p\ 

m which p denotes the arithmetical value of */a. 

This equation can be put under the form 

ic* _ p4 — ; 
which reduces to 

and tliis equation can be satisfied, by supposing 
/p2 _ ^2 _ Q . whence, x =z dc. p\ 
or by supposing 
a;2 



(.2 _|_^2 _ 0, whence, x = ± ^ —p"^ = ± ^ V — 1, 



CHAP. VII.] TRANSFORMATION OF RADICALS. 226 

We therefore obtain four different algebraic expressions for the 
fourth root of a. 

3. As another example, solve the equation 

X^ — p^ ■=: 0. 

This equation can be put under the form 

which may be satisfied by making either of the factors equal', 
to zero. 

But, a;3 — ^3 = 0, gives 



x=p, and x= p^ j. 

And if in the evquation a;3-f-j&3 — o, we make p=:^p'^ it. 
becomes x^ — p'^ = 0, from whicL we deduce 



x=p\ and x=:p'y j; 

or, substituting for p' its value — ^, 



X 



-P, and x=z-p\ ^ j. 



Therefore, x in the equation 

^6 _ ^6 _ Q^ 

and consequently, the 6th root of a, admits of six different algt- 
hraic expressions. If we make 

— , and a' = zr^ , 



2 
these expressions become 

p, ap, a'p, -^ p^^ ap,^ a'p. 

It may be demonstrated, generally, that there are as many 
different expressions for the n*^ root of a quantity as there are 
units in w. If 71 is an even number, and the quantity is posi- 
tive, two of the expressions will be real, and equal, with con- 
trary signs ; all the rest will be im&ginary : if the quantity is 
negative, they will all be imaginary. 

15 



226 * ELEMENTS OF ALGEBRA. LCH^lP. YIL 

If n is odd, one of the expressions will be real, and all the 
rest will be imaginary. 

161. If in the preceding article we make a = 1, we shall fmd 
the expressions for the second^ thirl^ fourth^ &c., roots of 1. 
Thus, + 1 and — 1 are the square roots of 1. 



Also, + 1, - ^ +/^ . and ^i^- 
are the cube roots of 1 : 



And +1, 1, -\-^ —I and — -y/ — 1, are the fourth 
roots of 1, &;c., &c. 

Rules for Imaginary Expressions. 

162« We shall now explain the modification of the ruies for 
operating upon radicals when applied to imaginary expressions. 

The product of ^ — a by ^ — a, by the^ rule of Art. 156, 
would be ^ -f- a^. Now, ^ + o? is equal to ± a, whence there 
is an apparent uncertainty as to the sign of a. The true pro- 
duct, hf ^ever, is — a, since, from the definition of the square 
root of a quantity, we have only to omit the radical sign, to 
obtain the quantity. 

Again, let it be required to form the product 



y-ax/-6. 
By the rule of Art. 156, we shall have 



but the true result is —J~ab^ so long as both the radicaU 



^ —a and ^ — h are afl^tcted with the sign + • 

For, ^ -a = ^, y ^T; and ,J^^ = ^-^ - 1 , 
heuce, 

- 1 = -.A^. 




CHAP. VII.J TRANSFORMATION OF RADICALS. 227 

In a similar manner, we treat all other imaginary expressions 
of the second degree ; that is, we first reduce them to the form 



of ay/ — 1, in which the co-efficient of -y/ — 1 is real, and then 
pi'oceed as indicated in the last article. 

162*. For convenience, in the application cf the preceding 
principle, we deduce the different powers of -y/ — 1, as follows: 






The fifth power is evidently the same as the first power ; the 
sixth power the same as the second ; the seventh the same as 
the third, and so on, indefinitely. 



163. If it is required to find the product of t/— a and 
*/ — 6, we should get, by applying the rule of Art. 156. 
1/ — a X \f~— ^ = X/ + ^^? but this is not the true result, 
For, placing the quantities under the form 

V^X\A^ and y6xl/=T, 
and proceeding* to form the product, we find 



a X 



since, {i/ — 1)^ = j\/ ^Z — 1 j = y' — 1 from the definition of 
a root. % 

Hence, generally, when we have to apply the rules for radi- 
eals to imaginary expressions of the fourth degree, transform 
them, so that the only factor under the radical sign shall be 
— 1, and then proceed as in the above example. 

Let us illustrate this remark, by showing cnat -^^ 

is an expression for the cube root of 1, or that, ir the restricted 
feense^ it is a cube root of 1. 



228 

We have 



ELE3IENTS OF ALGEBRA. 



I CHAP. VTT. 



'- -\-^r 



'h{- 



1 -\-^/^^^=^' 



\ 2 / \ 2 /' 

(-1)3 + 3. (-1)^.^3. V^ + 3.(-l).(V^)^(/T)^+(v/3)3.(vCr)3 



-14-3 -v/S". -v/^^nr -3 X — 3-3 V3. -yZ-l 



8 



= 1. 



In like manner, we may show, that 



1 



^-3 



is another 



expression for the cube root of 1, when understood in the 
restricted sense. It may be remarked that either of these ex- 
pressions is equal to the square of the other, as may easily 
be shown. 



Of Fi'adipnal and Negative Exponents. 



1- 



164. We have yet to explain a system of notation by means 
of which operations upon radical quantities may be greatly 
simplified. 

' We have seen, in order to extract the n^^ root of the quan- 
tity a"*, that when '//i is a multiple of ti, we have simply to 
divide the exponent of the power, by the index of the root to 
be extracted, thus, 

n / « 

When m is not a multiple of n, it has been agreed to 
retain the notation. 



or = a^^ 

these tsvo being regarded as equivalent expressions, and both 
indicating the 7i*^ root of the m^^ power of a, or what is the 
same thing, the m*^ pc Trer of the n*^ root of a; and generally, 

When any quantity ts written with a fractional exponent, the 
numerator of the fraction denotes the power to which the quantity 
is to he raised, and the denominator indicates the root of this 
power which is to he extracted. 



v.. 

CHAP. VII.] THEORY OF EXPONENTS. 229 

165. We have also seen that a^ may be divided by a", 

when m and n are whole numbers, by simply subtracting n 

from 771, giving 

a"* 

— = ft""-" = aP : 

a» ' 

in which we have designated the excess of m over n by p. 

Now, if n exceeds m, p becomes negative, and the exact 

division is impossible ; but it has been agreed to retain the 

notation 

a*" 

— = a"^" = arP, 

a" 

But when m <^n, in the fraction, 

a"* 

^' 

we may divide both terms by a'", and we have 

a*" 1 1 



a" a"~™ aP ' 

lience, a-P is equivalent to — , and both denote the recipro- 
cal of aP. 

We have, then, from these principles, the following equiva- 
lent expressions, viz. : 

^L/a equivalent to a". 



cr-". 







ya^ 


or 


{'■/^r 


u 






1 






a 






a« 












1 


or 


Vi 


(C 






1 


or 




(( 


166. 


It has 


been shown above that 



1 



a""" : if now we 



divide 1 by both members of this equation, we shall haA^e, 
a" = -^ : hence we conclude that, 



230 ELEMENTS OF ALGEBRA. [CHAP, VII 

Any factor may be transferred from the numerator to the de- 
nominator, or from the denominator to the numerator, by changing 
the sign of its exponent. 

167. It may easily be shown that the rules for operating 
upon quantities when the exponents are positive 'whole numbers, 
are equally applicable when they are fractional or negative. 

In the first place, it is plain that both numerator and 
denominator of the fractional exponent may be multiplied by 
the same quantity without altering the value of the expression, 
since by definition the m'* power of the m*^ root of a quan- 
tity is equal to the quantity itself. This principle enables us 
to reduce quantities, having fractional exponents, to equivalent 
ones having a common denominator. 

m r 

Let it be required to find the product of a" and a*' 

m r ms nr 

We have, a" X a« = a"* X a"''' 

or (Art. 164), "y^a'"* X «l/a^= ns^ms + nr' 

ws + w 

This last result is equivalent to a "* ' hence, 

■m r 77J5 + wr ^ 

a^ y^a} = a «« ' 

liie same result that would have been obtained by the appli- 
cation of the rule for the multiplication of monomials, when 
the exponents are positive whole numbers. 
If both exponents are negative, we shall have, 



-^--^11 1 


ms + nr 


TO r m3-\- n\ 




a" a« a "« 





If one of the exponents is positive, and the other negati-^e, 
ir» shall have, 

171 r m 1 ms 1 

a • X a" "^ = a~ X — = a »» X -— 

r nr » 

a* a^* 



CHAP VII.] THEORY OF EXPONENTS. 231 

whence, »y a'"* X y —;rr — V ~~^ — n5 /^mi-nr = 0^ «» » 



r m.s — nr 



and finally, a" x a * = a »* <J 

We have, therefore, for the multiplication of quantities when '"^ 
the exponents are negative or fractional, the same rule as when 
they are positive whole numbers, and consequently, the same 
rule for the formation of powers. 



EXAMPLES. 



3_1 23 lJi_2 

1. aH 2cr-i X a^'^c^ =a^ b^c ^ 

2. 3a- 2^,^ X 2a~^~6V = 6a~ ^ b^c\ 

3. Qa~^b*c-'^ X 5tt^6-V z= 80a ~ ^6 -'€"-« 



x4' Find the square of |a^. 



We have, (|a^')' = (J)^ X J""^ = |al 

5. Find the cube of Ja . Ans. ^a \ 

m r 

168. Let it be required to divide a" by a* . We shall have, 

m TO 

~ m r Z, ms — rn 

a" - - -' , * , ^^x a ** 

= a 



- zzra" X a * or (Art. 167), — 



a' a 



If both expon^ its are negative. 



rn — ms 



^ = a "^ X a* = a "» ' by the last article. 

If one exponent is negative, 

~ m r WIS + rn 

^ =a^ X a* z=i a "' ' by the precedmg article, 

/.~7 



232 ELEMENTS OF ALGEBRA. [CHAP. Vn. 

Hence, we see that the rule for the division of quantities, 
with fractional exponents, is the same as though the exponents 
were positive whole numbers ; and consequently we have the 
same rule for the extraction of roots, as when the exponents are 
positive whole numbers. 

EXAMPLES. 

3. 4 3._4 _ 1_ 

2 3 _X 1 _9_ _ 1 

4. Divide 2,2ah^c^ by ^a^h^c'^. Ans. Aa^hc*,\ 

5. Divide 64a96^c~^ by Z^a-^'^c"^ . Ans. 2a^^b^, 

3 /~2 2 

6. \/a^ =a^: 7. 



2 4 / 8 2 

.9 . <y . /..IT TT 



/~~^ _3 3 fl 1 _2 

8. x/a ^=za ^; 9. x/aH-^ = a^b '\ 

169. We see from the preceding discussion, that operations to 
be performed upon radicals, require no other rules than those 
previously established for quantities in which the exponents are 
entire. These operations are, therefore, reduced to simple oper 
ations upon fractions, with which we are already familiar. 

GENERAL EXAMPLES. 
, r, J 2n/2 X (3)^ . . , 

1. Keduce —^ to its simplest terms. 



Ans. 4^/a 



i(2)* 



2. Reduce -( ^7 ( to its simplest terms. 

(2i/2(3)* ) 



A... 1^ 1/3. 




THEORY OF EXPONENTS. 233 




/1\3_J_ /*^1-J * 

U; ~r V _1 ( ^Q ji-g sijYiplcst terms. 
2^.(l)M - 

4. What is the product of 

5 1,3 2 145 44- 

«2 ^ ^2^* ^ a\b\-{- ah + aH"" + 6^ by c? - b\ 

Ans. a^ — b^. 

Divide a^ — a^"'^-a^b■^Iy\ by a^ -~ 5"* 



^ "^ i. . ;i ... J ..r^ 



X 



170. If we have an exponent which is a decimal fraction, as, 
for example, in the expression 10 • ^oi from what has gone bfv 

_3JU_ 

fure the quantity is equal to (10)^°^°' or to iooo^/^o)^^S the 
value of which it would be impossible to compute, by any process 
yet given, but which will hereafter be shown to be nearly equal 
to 2. In like manner, if the exponent is a radical, as Vo, ^/TT, 
&;c., we may treat the expression as though the exponents were 
fractional since its values may be determined, to any degree of 
exactness, in decimal terms. ^ / j - 



CHAPTER VIL. 

OF SEHlEo AR THMETICAL PROGRESSION GEOMETRICAL PROPORTION AND 

PROGRESSION RECURRING SERIES BINOMIAL FORMULA SUMMATION OF 

SERIES PILING SHOT AND SHELLS. 

171. A SERIES, in algebra, consists of an infinite number of 
terms following one another, each of which is derived from 
one or more of the preceding ones by a fixed law. This law 

is called the law of the series. 

Arithmetical Progression. 

172. An ARITHMETICAL PROGRESSION is a serics, in which each 
terra is derived from the preceding one by the addition of a 
constant quantity called the common difference. 

If the common difference is positive, each term will be greater 
than the preceding one, and the progression is said to be in 
creasing. 

If the common difference is negative, each term will be less 
than the preceding one, and the progression is said to be 
decreasing. 

Thus, ... 1, 3, 5, 7, . . . &;c., is an increasing arithmetical 
progression, in which the common difference is 2 ; 

and 19, 16, 13, 10, 7, ... is a decreasing arithmetical 

progression, in wliich the common difference is — 3. 

173. When a certain number of terms of an arithmetical 
progression are considered, the first of these is called the Jirst 
term of the progression, tha last is called the last term of the 
progression, and both together are called the extremes. All the 
terms between the extremes are called arithmetical means. Au 
arithmetical progression is often called a progression by differences. 



i 

CHAP. Vni.] ARITHMETICAL PROGRESSION". 235 

174. Let d represent the common difference of the Arithmeti- '''- 
cal progression, 

a.b.c.e,f.g.h.1c^ &c., 
which is written by placing a period between each two of the 
terms. 

From the definition of a progression, it follows that, 
b z= a -\- d^ c =:h -{- d := a-^ 2d, e=:c-\-dz=a-{-Sd\ 
iijd, in general, any term of the series, is equal to the first 
term plus as many times the common difference as there are pre- 
ceding terms. 

Thus, let / be any term, and n the number which marks the 
^lace of it. Then, the number of preceding terms will be de- 
noted by 71 — 1, and the expression for this general term, will be 
I ^=z a -\- {ill — X) d. 
If d is positive, the progression will be increasing ; hence, 

\ In an increasing arithmetical progression, any term is equal to 
Ihe first term, plus the product of the common difference by the 
number of preceding terms. 

If we make w = 1, we have / = a ; that is, there will be 
but one term. 

If we make 

71 = 2, we have l = a -\- d-, 
that is, there will be two terms, and the second term is equal 
to the first plus the common difference. 

EXAMPLES, 

V 

1. If a = 3 and d — 2, what is the 8d term? Ans, 7. 

2. If a = 5 and c? =: 4, what is the 6th term ? Ans. 25. 

3. If a = l and d—f). what is the 9th term ? Ans, 47. 

The formula, 

l = a^{n-\)d, 
serves to find any term whatever, without determining those 
M^hich precede it. 



236 ELEMENTS OF ALGEBSA. [CIIAP. TIIL 

Tims to find the 50th term of the progresskn, 
1 . 4 . 7 . 10 . 13 . 16 . 19, . . 

we have, ^ = 1 + 49 X 3 = 148. 

And for the 60th term of the progression, 

1 . 5 . 9. 13 . 17 . 21 . 25, . . . 

we have, / = 1 + 59 x 4 = 237. 

174*. If d is negative, the progression is decreasing, and the 
formula becomes 

? = a — (w, — 1)(/; that is. 

Any term of a decreasing arithmetical progression^ is equal to 
the first term, plus the product of the common difference by the* 
number of preceding terms. 

EXAMPLES. 

1. The first term of a decreasing progression is 60, and the 
common difference — 3 : what is the 20th term 1 

l = a-{n-l)d gives / =. 60 - (20 - 1) 3 = 60 - 57 = 3. 

2. The first term is 90, the common difference — 4 : what 
js the 15th terml Ans. 34. 

3. The first term is 100, and the common diflference _ — 2* 
what is the 40th term? Ans. 22. 

175. If we take an arithmetical progression, < : 

a . b . c i . k . I, 

having n terms, and the common diflference d, and designate 
the term which has p terms before it, by t, we shall have 
t — a-{-pd . . - - - (1). 
If we revert the order of terms of the progression, con- 
Bidering I as the first term, we shall have a new progression 
whose common difference is — d. The term of this new pro- 
gression which has p terras before it, will evidently be the same 
as that which has p terms after it in the given progression, 
and if we represent that term by t\ we shall have, 
t' = l-pd - ... (2). 



CHAP. Vlir.J ARIIHAIETICAL PROGRESSION. 237 

Adding equations (1) and (2)^ member to member, we find 
i-\- t' — a-\-l \ hence, 

The' sum of any two terms^ at equal distances from the extremes 
of an aritJimetical progression^ is equal to the sum of the extremes, 

176. If the sum of the terms of a progression be repre- 
sented by >S', and a new progression be formed, by reversing 
tlie order of the terms, we shall have 

Sz^a + h + c^ . . . . 4-i + ^-f-^, 
Sz^l +k-^i+ . . . . +c+5 + a. 
Adding these' equations, member to member, we get 
2,^.= («+/)+ (6 + ^)+(c + ^) . . . +{i-\-c) + {1c-\-h) + {l+a)', 

and, since all the sums, a + ^, h -\- k^ c -{- i . . . . are equal 
to each other, and their number equal to n, the number of 
terms in the progression, we have 

2S = {a -{- I) n^ or ^ —\ — -^ — I ^ j ^^^^^ i^j 

The sum of the terms of an arithmetical progression is equal to 
half the sum of the two extremes multiplied by the number of terms. 

EXAMPLES. 

1. The extremes are 2 and 16, and the number of terms 8: 
n^hat is the sum of the series ? 

<a + Z\ . ^2+16 



.S' 



= ^— - j X n, gives S = — ^— X 8 = 72. 



2. The extremes are 3 and 27, and the number of terms 12 •, 
what is the sum of the series'? Ans. 180. ( 

3. The extremes are 4 and 20, and the number of terms 10: 
what is the sum of the series ? Ans. 120. 

4. The extremes arc 8 and 80, and the number of terms 10: 
what is the sum of the series 1 Ans. 440. 

The formulas 

l = a + {n-\)d and S={~^\xn, 



238 



ELEMENTS OF ALGEBRA. 



[CHAP. VIIL 



contain five quantities, a, c?, w, /, and *S', anc consequently give 
rise to the following general problem, viz. : 

Any three of these Jive quantities being given^ to determine the 
other two. 

This general problem gives rise to the ten following cases :— 



Given. 



a, d. n 



I'nknown.i 



Values of the Unknown Quantities. 



/, *S Jr=a ^- (;i - !)(/; S :^\ n \2a ^ (71 — \) d\ 



a, c/, / 



w, >S' 






a^d.S 



n, I 



d-2u±,/{cr^2af-\-8dS 



2d 



; Izzza-^ {71 — l)d. 



S, d. 



^=i?i (« + /); d 



i — a 



a, n^ S 



d, I 



d= 



2{S 



2S 



n{n- 1) 






a, L S 



n, d 



2S ^ (Z4-«)(/-a) 



d, n, I 



«, s 



a=l-{n-l)d- S = in[2l-{n- \) d]. 



d,n,S 



a. I 



2S — n{n — l)d^ 2S -^ n {n — I) d 

2n ' ~ 2^n * 



?i, a 



2l^d^y^{2[-\-df-8dS 



2d 



; a ziz I — (n — \)d. 



10 



71, Z, S, a, d 



n ' w (m — 1) ' 



177. From the formula 

I — a -{- {n — \)d, 
we have, a =z I — (n — 1) ^ *, that is, 

The first term of an increasing arithmetical progression^ is eqital 

to any following term^ minus the product of the coirjnon difference 

by the number of preceding terms. 

178. From the same formula, we also find 

I —a . 

d — ; that is, 

n — i 



CHAP. VIII.J ARITHMETICAL PKOaEESSION. 239 

In any arithmetical progression^ the common dijference is equal 
to the lust term minus the first term^ divided by the number of 
terms less one. 

If the last term is less than the first, the common difTerence 
will be negative, as it should be. 

EXAMPLES. 

I 

1. The first term of a progression is 4 the last term 16, and 

the nuMilv^r of terms considered 5 : what is the common 
difference 1 

The formula 

, /-a . , 16-4 „ 

d = ffives d = = 3. 

n — 1 ° 4 

2. The first term of a progression is 22, the last term 4, 
and the number of terms considered 10 : what is the common 
difference ? Ans. — 2. 

179. By the aid of the last principle deduced, we can soivo 
the following problem, viz. : 

To find a. number m of arithmetical means between two given 
numbers a and b. 

To solve this problem, it is first necessary to find the corn- 
inon diflTerence. Now, we may regard a as the first term of 
an arithmetical progression, b as the last term, and the required 
means as intermediate terms. The number of terms considered, 
of this progression, will be expressed by m -f- 2. 

Now, b} substituting in the above formula, b for /, and m -f 2 
for /I, it becomes » 

- b — a . b — a 

d = r, or a = 



m + 2-1' m+r 

that is, the common difference of the required progression Is 
obtaired by dividing the difference between the last and first 
terms by one more than the required number of means. 



240 ELEMENTS OF ALGEBRA. [CHAP. VIIL 

Having obtained the common difference, form the second term 
of the progression, or the first arithmetical mean, hj adding c?, or 

-. to the first term a. The second mean is obtained by 

VI -f- r '' 

augmenting the first by d, 6zq, 

EXAMPLES. 

1. Find 3 arithmetical means between 2 and 18. The formula 

d = — — — , gives d = = 4 ; 

m+1' ^ 4 ' ^ 

hence, the progression is 

2 . 6 . 10 . 14 . 18. 

2. Find 12 arithmetical means between 77 and 12. The 
formula 

b -a . ,12-77 

f 
hence, the progression is 

77 . 72 . 67 . 62 22 . 17 . 12. 

8. Find 9 arithmetical means and the series, between 75 
and 5. 

Ans. Progression 75 . 68 . 61 26 . 19 . 12 . 5. 

180. If the same number of arithmetical means be inserted 
between the terms of a progression, taken two and two, these 
terms, and the arithmetical means together, will form one and 
the same progression. 

For, let a . 6 . c . e . / . . . . be the proposed progression, 
and m the number of means to be inserted between a and 6, 
b and c, c and e 

From what has just been said, the common difference of 
each partial progression will be expressed by 

b — a c — b e — c 
m+ r m+ 1' m-f 1 * * * * 

which are equal to each other, since, a, 6, c, . . . are in pro 
gression : therefore, the common difference is the same in each 



CHAP. VIII.J ARITHMETICAL PROGRESSION. 241 

of the partial progressions; and since the last term of tiie first, 
forms the first terra of the second, tSic, we may conclude that 
all of these partial progressions form a single progression. 

GENERAL EXAMPLES. 

1. Find the sum of the first fifty terms of the progression 

2.9. 16.23 . .. 
For the 50th term, we have 

Z == 2 + 49 X 7 = 345. 

Hence, S={2^ 345) x ^ = 347 x 25 = 8675. 

2. Find the 100th term of the series 2 . 9 . 16 . 23 . . 

Ans. 695. 

3. Find the sum of 100 terms of the series 1.3.5.7.9..,. 

Ans. 10000. 

4. ITie greatest term considered is 70, the common difierence 
3, and the number of terms 21 : what is the least term and 
the sum of the terms'? 

Ans. Least term 10; sum of terms 840. 

5. The first term of a decreasing arithmetical progression is 
10, the common difierence is — ^, and the number of terms 
21 : required the sum of the terms. Ans. 140. 

6. In a progression by difierences, having given the common 
difference 6, the last term 185, and the sum of the terms 2945 : 
fmd the first term, and the number of terms. 

Ans. First term —b\ number of terms 31. 

7. Find 9 arithmetical means between each antecedent attvi 
consequent of the progression 2. 5. 8. 11. 14... 

Ans, c? = 0.3. 

8. Find the number of men c^tained in a triangular bat- 
talion, the first rank containing 1 man, the second 2, the third 
3, and so on to the n*^^ which contains n. In other words, 

16 



242 ELEMENTS OF ALGEBRA. [CHAP. VIII. 

find the expression for the sum of the natural numbers 1, 2, 
8, . . . from 1 to n. inclusively. , , _. 

Ans, S = -. 

0. rind the sum of the first n teriis of the progression of 
uneven numbers 1, 3^ 5, 7, 9 . . . Ans. S =z n\ 

10, One hundred stones being placed on the ground, in a 
-straight line, at the distance of two yards from each other, ho\r 
far will a person travel who shall bring them one by one to 
a basket, placed at two yards from the first stone 1 

Ans. 11 miles 840 yards. 

Of Ratio and Geometrical Proportion. 

181. The Ratio of one quantity to another, is the quotient 
which arises from dividing the second by the first. Thus, the 

ratio of a to h, is — . 

182. Two quantities are said to be proportional, or in pro- 
portion, when their ratio remains the .same, w^hile the quantities 
themselves undergo changes of value. Thus, if the ratio of a 
to 6 remains the same, while a and b undergo changes of value, 
then a is said to be proportional to h. 

183» Four quantities are in proportion^ when the ratio of the 

first to the second, is equal to the ratio of the third to the 

fourth. 

Tims, if 

h__d_ 

a ~~ c ^ 

the quantities a, b, c and c?, are said to be in propoition. We 
generally express that these quantities are proportional by wrilu g 
them as follows : 

a : b : : c : d. 

This algebraic expression is read, a is to 6, as c is to i, 
and is called a proportion. 



CHAP. VIII.] GEOMETRICAL PROGRESSION 245 

184. Tlie quantities compared, are called terms of the pro- 
per tioti. 

The nrst and fourth terns are called the extremes^ the scconi 
and third are called the rneans ; the first and third are called 
onteredents, the second and fourth are called consequents^ and the 
fourth is said to be a fourth proportional to the other three. 

If the second and third terms are the same, either of these 
is said to be a mean proportional between the other two. Thus, 
in the proportion 

a : b : : 6 : c, 
6 is a mean proportional between a and c, and c is said to be 
a third proportional to a and b. 

185i Two quantities are reciprocally proportional when one is 
proportional to the reciprocal of the other. 

Geometrical Progression. 

186. A Geometrical Progression is a series of terms, each 
of which is derived from the preceding one, by multiplying it 
\y a constant quantity, called the ratio of the progression. 

If the ratio is greater than 1, each term is greater than u\e 
preceding one, and the progression is said to be increasing.. 

If the ratio is less than 1, each term is less than the pr*^' 
ceding one, and the progression is said to be decreasing. 

Thus, 
... 3, 6, 12, 24, . . . &;c., is an increasing progression. 

. . . IG, 8, 4, 2, 1, — , — , . . . is a decreasing progressl^^ 

It may be observed that a geometrical progression is a cori- 
tinued proportion in which each term is a mean proportions^ 
between the preceding and succeeding terms. 

187« Let r designate the ratio of a geometrical progression, 

a : b : c : d, . . . . &c. 
We deduce from the defmition of a progression the follow 
ing equations : 

b = ar^ c zzzbr = ar^, d = cr = ar^^ e = dr =. aa** , . ; 



244 ELE31ENTS OF ALGEBRA. [CHAP. VIIL 

and, In general, any term w, that is, one which has n — 1 terms 
before it, is expressed hy ar'^^. 

Let I be this term ; we have the formula 
I = ar'^\ 
by means of which we can obtain any term without being 
obliged to find all the terms which precede it. That is, 

. Any term of a geometrical progression is equal to the Jirst term 
multiplied by the ratio raised to a power whose exponent denotes 
the number of preceding terms. 

/ EXAMPLES. 

1. Find the 5th term of the progression 

2 : 4 : 8 : 16, &c., 
in which the first term is 2, and the common ratio 2. 
5th term = 2 X 2* = 2 X 16 = 82. 

2. Find the 8th term of the progression 

2 : 6 : 18 : 54 . . . 
8th term = 2 X 3' = 2 X 2187 = 4374. , 

3. Find the 12th term of the progression 

64 : 16 : 4 : 1 : 4- • . 
4 

/ 1 \^i 43 1 - 1 
12th term = 64 (-) = _ = _ = ^^. 

188. We will now explain the method of determhiing the sum 
of n terms of the progression 

a '. h : c '. d '. e : f \ . , , : i : k : ly 

of which the ratio is r. 

If we denote the sum of the series by S^ and the ■«'* term 
by Z; we shall have 

S =: a-{- ar -\- ar"^ . . . . + ar'*"^ -j- ar*"*. 
If vre multiply b)th members by r, we have 

Sr zzzar -\- ar^ -\- ar^ , , . -\- a;**-- -|- «»*" ; 



CHAP. VIII.l GEOMETRICAL PROGRESSION. 245 

and by subtracting the first equation fr^m the second, member 
from member, 

^ ar"^ — a 

Sr — S = ar'' — a, whence, S = ; 

r — 1 

substituting for a?-", its value /r, we have 

7i» g^ 

jS — • that is, 

r — 1 

To obtain the sum of any number of terms of a progressioa 
by quotients, 

Multiphj the last term hy the ratlo^ siihtract the first term from 
tJiis product^ and divide the remainder by the ratio diminisJted by 1. 

EXAMPLES. 

1. Find the sum of eight terms of the progression 

2 : 6 : 18 : 54 : 162 . . . : 4374. 

]r — a 1,^122 — 2 

Sz^- ? := ±^— ^ = 65G0. 

r — \ 2 

2. Find the sum of five terms of the progression 

2 : 4 : 8 : 16 : 32 ; . . . . 

^^^^^i^^62. 
?' — 1 1 

3. Find the sum of ten terms of the progression 

2 : 6 : 18 : 54 : 162 . . . 2 X 39 z= 39366. 

Ans. 59048. 

4. What debt may be discharged in a year, or twelve months, 
by paying %1 the first month, $2 the second month, 84 the third 
month, and so on, each succeeding payment being double the 
k',st ; and what will be the last payment % 

Ans. Debt, $4095 ; last payment, $2048. 

5. A gentleman married his daughter on New- Year's day, and 
gave her husband \s, toward her portion, and was to double it 
on the first day of every month during the year : --vhat was hei 
portion? Ans. £204 15*. 



246 ELEMENTS OF ALGEBRA. [CHAP. VIIL 

6. A man bought 10 bushels of wheat on the condition that 
he should pay 1 cent for the first bushel, 3 for the second, 9 
for the third, and so on to the last : what did he pay for 
the last bushel, and for the ten bushels 1 

Arts. Last bushel, $196 S3; total cost, $295,24. 

189. When the progression is decreasing, we have r < 1 and 

/ < a ; the above formula for the sum is then written under 

the form 

a — lr 

in order that both terms of the fraction may be positive. 
By substituting ar"-^ for I, in the expression for >S', 

ar" — a ^ a — ar^ 

S = 1-, or 0=-; . 

r— 1 1 — r 

EXAMPLES. 

1. Find the sum of the first five terms of the progression 

32 : 16 : 8 : 4 : 2. 

32-2x4- oi 

1 — r 1 1 

Y "2" 

2. Find the sum of the first twelve terms of the progression 
64 : 16 ; 4 : 1 : -i- ^ 



4 65536 

_ q. - Ir _ ^^ "" 65536 ^ T _ ^^^ ~ 65536 _ 65535 



l-r _3_ 3 196608* 

4 

We perceive that the principal difficulty consists in obtauihig 
the numerical value of the last term, a tedious operation, even 
whnn the number of terms is not very great. 

190. If in the formula 

^^' r-l ' 



CHAP. VIII.l GEOMETRICAL PROGRESSION". 247 

we make r — l, it reduces to 

^- 0- 

Tliis result sometimes indicates indetermination ; but it ofteu 
wises from the existence of a con^mon flictor in both numerator 
and denominator of the fraction, which factor becomes 0, in coi^- 
sequence of a particular supposition. 

Such is the fact in the present case, since both terms of the 
fi'action contain the factor r — 1, which becomes 0, for the par- 
ticular supposition r = 1. 

If we divide both terms of the fraction by this common factor, 
we shall find (Art. 60), 

,S^ z= ar"-'^ + ar«-2 + ar"-3 -f .... -{. ar -{- a, 

in which, if we make r — 1, we get 

Sz=za-\-a-{-a-\-a-{- -f-«= na. 

We ought to have obtained this result; for, under the suppi>. 
eition made, each term of the progression became equal to a, 
ard since there are n of them, their sum should be na. 

191. From the two formulas 

rv ^^ — ct 
I = ar"^-^, and S — 



r -V 

several properties may be deduced. We shall consider only 
some of the most important. 
The first formula gives 

I n— 1 /"/" 

j.n-1 — — whence r= \/-—. 
a V a 

The expression 

n~i rj 



furnishes the means for resolving the following problem, viz . 

To find m geometrical means between two given numbers a and 
b ; that is, to find a number m of means, which will form vnth a 
and i», considered as extremes, a geometrical progression. 



24:8 ELEMENTS OF ALGEBRA. | CHAP VIII, 

To find this series, it is only necessary to know the ratio. 
Now, the required number of means being m, the total number 
of terms considered, will be equal to w -\ 2. Moreover, we 
have / = 6 ; therefore, the value of r becomes 

r = \ / — ; that is, 

To find the rat'io^ divide ike second of the given numbers by the 
first; then extract that root of the quotient whose index is one 
greater than the required number of means : 
Plence the progression is 

m+\ f~^ 771+1 n^ ^+1 



771 + 1 / 



a : a \/ — : a \/ — :: : a 
a 



EXAMPLES. 



1. To insert six geometrical means between the numbers 3 
and 884, Ave make m = 6, whence from the formula, 

7 /SS4 



.= yi28 = 



2: 



o 

hence, we deduce the progression 

3 : 6 : 12 : 24 : 48 : 96 : 192 : 384. 
2. Insert four geometrical means between the numbers 2 and 
486. The progression is 

2 : 6 : 18 : 54 : 162 : 486. 

Remark. — When the same number of geometrical means are 
inserted between each two of the terms of a geometrical pro- 
gression, all the progressions thus formed will, when .,aken to- 
gether, constitute a single progression. 

Progressions having an infinite number of terms. 

192» Let there be the decreasing progression 
a '. b '. c \ d '. e \ f '. , . ., 
etmtaining an infinite number of terms. The formula 



CHAP. VIII.] GEOMETRICAL PROGRESSIOxNT. 249 

which expresses the sum of n terms, can be put under the form 

a ar^ 



I — r I — r 

Now, since the progression is decreasing, r is a proper frac- 
tion, and r" is also a fraction, which diminishes as n increases. 
Therefore, the greater the number of teims we take, the more 

will X r" diminish, and consequently, the nearer will the 

sum of these terms approxiniate to an equality with the first 

part of S] that is, to . Finally, when n is taKcn greater 

than any assignable number, or when 

n =z CO, then X r^ 

\ — r 

will be less than any assignable number, or will become equal 
to ; and the expression will represent the true value of 

the sum of all the terms of the series. Hence, 

The sum of the terms of a decreasing progression^ in which the 
number of terms is ivjinite, is 

a 



S 



This is, properly speaking, the limit to which the partial sums 
approach, as we take a greater number of terms of the pro- 
gression. The number of terms may be taken so great as to 

make the diflerence between the sum, and , as small as 

1 — r 

we please, and the difference will wly become zero \ l.on the 

number of terms taken is infinite. 



EXAMPLES. 



1. Find the sum of 



1-1 .11 — & 



250 ELEMENTS OF ALGEBRA. [CHAP. Vm. 

We have, for the sum of the terms, 

a 1 3 



S = 



1 - r 1^ 2 

"3 




2, Again, take the progression 

1 1 1 ^ ^ 1 ^ 

We have S=-^— =■■ — ^ = 2. 

What is the error, in each example for ?i = 4, r. = 5, ri == 6 1 

Indeterminate Co-efficients. 

193. An Identical Equation is one which is satisfied for any 
values that may be assigned to one Or more of the quantities 
which enter it. It differs materially from an ordinary equation. 

The latter, when it contains but one unknown quantity, can 
only be satisfied for a limited number of values of that quan- 
tity, whilst the former is satisfied for any value whatever of 
the indeterminate quantity M'hich enters it. 

It differs also from the indeterminate equation. Thus, if in 

the ordinary equation 

ax -\- b]/ -\- cz -\- d =1 

values be assigned to x and y at pleasure, and corresponding 
values of z be deduced from the equation, these values taken 
together will satisfy the equation, and an infinite number of 
sets of values may be found which will satisfy it (Art. 88). 
But if in the equation 

ax + b?/ -\- cz -\- d z= 0, 
we Impose the condition that it shall be satisfied for any 
valies of X, y and 0, taken at pleasure, it is then called an 
identical equation. 

194. A quantity is indeterminate when it admits of an infinite 
number of values. 

Let us assume the identical equati")n, 

A-\-Bx^' Oz'- + Dx^+ &LQ = 1 - - - - (1), 



CHAP. VIII.] GEOMETRICAL PROGRESSION". 251 

in which the co-efficients, A, B, C, i>, &c., are entirely inde 
pendent of x. 

If we make a; = in equation (1) all the term? contauiing 
9 reduce to 0, and we find 

A =0. 
Substituting this value of A in equation (1), and factoring, 
it becomes, 

x{B-{- Cx-\- Bx^ + &c.,) = (2), 

which may be satisfied by placing x = 0, or by placing 

B-]- Cx + Bx"^ -^ &LG. = (3). 

The first supposition gives a common equation, satisfied only 
for X = 0. Hence, equation (2) can only be an identical equa- 
tion undor a supposition which makes equation (3) an identical 
equation. 

If, now, we make x =z in equation (3), all the terms con- 
taining X will reduce to 0, and we fmd 

^ = 0. 
Substituting this value of B in equation (3), and factoring, 
we get 

x{C+Bx+ &c.) = (4). 

In the same manner as before, we may show that C = 0, 
and so we may prove in succession that each of the co-eflicienta 
Z), B, &c., is separately equal to : hence, 

In every identical equation, either membei' of which is 0, in- 
volving a single indeterminate quantity^ the co-efficients of the 
different powers of this quantity are separately equal to 0. 

195* Let us next assume the identical equation 

a -\- hx -{- r.x'2 -f (ScC. = a' -^ h'x -{- c'x"^ -f- &c. 
By transposing all the terms into the first member, it may 
be pJac'id under the form 

(a — a') + {h — h') x-\- {c— r') x"^ + &c. = 0. 
Now, from the principle just demonstrated, 

a — a' =r 0, b—b' — O, c — c' = 0, &c., &c. ; 
whence a = a' , b =z b' c = c' , &c., &c. ; that is, 



252 ELEMENTS OF ALGEBRA. LCHAP. YIU. 

In an identical equation containing but one indeterminate quan- 
tity, the co-efficients of the like powers of that quantity in the 
two members, are equal to each other. 

196. We may extend the principles just deduced to Identical 
equations containing any number of indeterminate CLuantities, 

"For, let us assume that the equation 
a -f- bz -i- b'y + b"z -\- &c. + cx^ + c'y'^ + c"z^ + &c. + dx^ 
+ 6?y + &c. = - - - (1), 
IS satisfied independently of any values that may be assigned 
to a:, y, 0, &c. If we make all the indeterminate "quantities 
xcept X equal to 0, equation (1), will reduce to 

a -\- bx -\- cx^ -\- dx^ + &c. =: ; 
whence, from the principle of article 191, 

a — 0, b = 0, = 0, d = 0, &c. 

If, now, we make all the arbitrary quantities except y equal 
to 0, equation (1) reduces to, 

a -^b'y + c'?/2 + d'y^ -f &c. =z ; 
whence, as before, 

a = 0, b' = 0, c' = 0, d' = 0, &c. 
and similarly we have 

b" = f , c" = 0, d:c. 

The principle here developed is called the pri?iciple of inde 
terminate co-efficients, not because the co-efficients are really 
indeterminate, for we have shown that they are separately 
equal to 0, but because they are co-efficients of indeterminate 
quani ities. 

197. The principle of Indeterminate Co-efficients is much used 
ir. developing algebraic expressions into ser es. 

For example, let us endeavor to develop the expression, 

a 

a -j- b'x^ 

ijato a series arranged according to the ascending powers of x» 



CE VP. VIII.] GEOMETRICAL PROGRESSION. 253 

Let us assume a development of the proposed form, 

-r^-TT- =P-{-Qx + Ex^ + Sx^ + &c. ... (1), 
a -\- b X 

in which P, .^, B, &c., are independent of x, and depend upon 

o, a' and b' for their values. It is now required to find such 

vulues for P, Q, i?, &c., as will make the development a true 

one for all values of x. 

By clearing of fractions and transposing all the terms into 
Jie first member, we have 

Pa' + Qa' x + Ra' x"^ + &c. = 0. 
— a -\- Pb' -\- Qb' &c. 

Since this equation is true for all values of a:, it is identi- 
cal, and from the principle of Art. 194, we have 
Pa' - a =n 0, Qa' + Pb' = 0, Pa' + Qb' = 0, &c., &c. ; whence, 

^ a ^ Pb' ab' ^ Qb' ab"^ , 

a a a'^ a a •^ 

Substituting these values of P, Q^ P, &c., in equation (1), 
it becomes 



— ^^ + -7^^ ;xa;3-f-(5r^ - - (2). 



a' -j-b'x a' a"^ a''^ a 

Since we may pursue the same course of reasoning upon 
any like expression, we have for developing an algebraic ex- 
pression into a series, the following 

RULE. 

I. Place the given expression equal to a development of the 
form P -\- Qx -{- Px^ -{- <&€., clear the resulting equation of frac- 
tions, and transpose all of the tcrm.s into the first member of 
the equation. 

II. Then place the co-efficients of the different powers c/ the let* 
ter, with reference to which the series is arranged, separately equal 
to 0, and from these equations find the values of P, Q, P, dbc. 

III. Having found these values, substitute them for P, Q, R, <&c., 
in the assumed develojjment, and the result will be the develop* 
ment required » 



254 ELEMENTS OF ALGEBRA. [CHAP. VIIL 



EXAMPLES. 



1. ])evelop into a series. 

a — X 

X x^ x^ 

Ans. 1 -f- 1- -I- _ -f _- + &0. 



*J. Develop -, — iiito a series. 

^ yd — xy 



Alls. -^ + -^ + — tH r-f&8. 

a^ a^ a* aP 



^ ^ , 1 4- 2.r . 

3. Develop -— mto a series. 

1 — o^r 

Ans. 1 + 5^; + Idx^ + 45^;^ -f lS5x* + &c. 

198. We have hitherto supposed the series to be arranged 
according to the ascending powers of the unknown quantity, 
commencing with the power, but all expressions cannot bts 
developed according to this law. In such cases, the application 
of the rule gives rise to some absurdity. 

For example, if we apply the rule to develop -, we 

tjX "~~ X 

shall have, 

1 



Sx 



P-h Qx + Bx^ + &c. - - - (1). 



Clearing of fractions, and transposing, 

— l-{-SFx-^SQ a:2+&c. =0; 
- F 
Whence, by the rule, 

-1=0, 8P = 0, 3^-P = 0, &c. 

Now, the first equation is absurd, since — 1 cannot equal 0. 
Hence, we conclude that the expression cannot be developed ac- 
cording to the ascending powers of x, beginning at x^. 

We may, however, write the expression under the forrc 

— X ^ , and by the application of the rule, develop the facte 

X o X 

, which gives 

C -' X 



CHAP. VIII.J RECURRING SERIES. 255 



whence, by substitution, 



1 a;0 J y, A ^2 1 <^c 



Zz-x^ 3a; ' 9 27 '81 

Since — is equal to 3r-' (Art. 166), we see that the true devel- 

opment contains a term with a negative expnnent, and the sup- 
position made in equation (1) ought to have failed. 

Recurring Series. 

199. The development of fractions of the form — --, &c., 

a-\-bx 

gives rise to the consideration of a kind of series, called recur- 
ring series. 

A RECURRING SERIES is ouc in which any term is equal to the 
algebraic sum of the products obtained by multiplying one or 
more of the preceding terms by certain fixed quantities. 

These fixed quantities, taken in their proper order, constitute 
what is called the scale of the series. 

200. H we examine the development 

a a ah' ah'"^ „ ah'^ „ 

a -\- X a' a ^ a ■^ a* 

we shall see, that each term is formed by multiplying the pre- 

h' . . 

ceding one by jx. This is called a recurring series of the 

Jirst order ^ because the scale of the series contains but one 
term. 

The expression -a* is the scale of the series^ and the ex 

b' 

pression j- is called the scale of the co-efficients. 

It may be remarked, that a geometrical progression is a recur 
ring series of the first order. 

201. Let it be required to develop the expression 

a + hx 



t! -j- h'x -f- c'x^ 



into a series. 



256 



Assume 



ELEMENTS OF ALGEBRA, 
a -\- bx 



[CHAP. VIIL 



a' + ^'^ + c'-^"^ 
Qearing of fractions, and transposing, we get 



Fa' 


+ Qa' 


x + Ra' 


x^ + Sa' 


— a 


+ py 


+ Qh' 


-\-Rb' 




-b 


+ Pc' 


■\-qc' 



x^ + &c. = 0. 



Therefore, we have 

Fa' — a = 0, 

Qa' -j- Fb' - b=zO, 

Ra' + Qy-{-F(/^0, 

Sa' + Rb' + Qc'=0, 

&c., &c., 

from which we see that, commencing at the third, each co-elTi- 

cient is formed by multiplying the two which precede it, re* 

y (/ 
spectively, by j and j, viz., that which immediately 

h' 
precedes the required co-efficient by ^, that which precedes 

(/ 
it two terms by j, and taking the algebraic sum of the pro 



or, 


F = 


a'' 


or, 


Q^ 


-i--i: 


or, 


R=z 


-^^-i^: 


or, 


S = 


&c., &;c.; 



ducts. Hence, 



(-- --] 

\ a'' a') 



is the scale of the co-efficients. 

From this law of formation of the co-efficients, it follows that 

the third term, and every succeeding one, is formed by multi- 

b' 
■ plying the one that next precedes it by jX^ and the second 

precedmg one by ; x"^. and then taking the algebraic sum of 



these products : hence, 



{-i^^' -^^') 



ia the scales of the series. 



CHA1». Vlll.J BINOMIAL THEOKEM. 257 

Tills scale contains two terms, and the series is called a re- 
curring series of the second order. In general, the order of a 
recurring series is denoted by the number of terms in the scale 
of the series. 

The development of the fraction 

a -\- hx -{• cx^ 
a' -\-b'x-{-&x'^ + d'x^' 

gives rise to a recurring series of the third order, the scale of 
which is, 






and, in general, the development of 

a -\- hx -{- cx"^ -\- . . . Jcx^-^ 
a' -\-h'x-\-C'x'^-^ . . . k'x^ ' 

gives a recurring series of the n*^ order, the scale of wliich is 
I re, X"^ , . , X^ I 

General demonstration of the Binomial Theorem. 

202. It has been shown (Art. 60), that any expression of the 
form z^ — ?/"*, is exactly divisible by z —y, when m is a po«^*^ivf. 
whole number, giving, 

2^fn nitn 

— = z^-^ + z'^-hj -f z''^^y^ + .... 4- y"^^ 

The number of terms in the quotient is equal to m, and if 
we suppose z ■= y, each term will become z^~^ ; hence, 



(z^ — y^\ 
z — y )v=r 



y /y 

The notation employed in the first member, simply indicates 
what the quantity within the parenthesis becomes when we make 
y =z. 

We now propose to show that this form is true when m is 
fractional and when it is negative. 

17 



258 ELEMENTS OF ALGEBRA. [CHAP. VIIL 

P 

First^ suppose m fractional, and equal to — . 

Make 2 ? = v, whence z^ =vp and s =:vi; 

JL I. 

and y9=u, whence y 2 z=: uP and y =u9 , 



hence, 



p p 


VP — 


vP — uP 


z9 —yi 


uP V — u 



Z — y v^ — W? V^ — Ui 

V — u 
If now, we suppose y = 2, we have v z= u, and since p and q 
are positive whole numbers, we have 

)zi-y9l _ \v-u Jv = u__ pvP^^p_^^^^p_^f-i 

qvi—^ q q 



z ivi — un 

\v — u }^^, 



y 

'^Second, suppose m negative, and either entire or fractional. 
By observing that 

— z-^ y-^ X {z^ — y^) = zr^ — y~*". 



we have, 

g— »» — y—n 



_ _ ^-my-^ X 



z — y z —y 

If, now, we make the supposition that y =^ z, the first factoi 
of the second member reduces to — z-"^^, and the second fac- 
tor, from the principles just demonstrated, reduces to m^"*-- ; 
hence, 

\ z-y Jy^^ 
We conclude, therefore, that the form is general. 

203i By the aid of the principles demonstrated in the last 
article, we are able to deduce a formula for the develop- 
ment of 

(x + «)"», 

t^'hen the exponent m is positive or negative, entire or fractional 
Let us assume the equation, 

il-\-z)^ = F+ Qz-{- Bz^ + Sz^ 4- &c. • - . (1), 



CHAP. VIII.J BINOMIAL THEOREM. 25^ 

iu which, P, (2, i2, &;c., are independent of z^ and depend upon 
1 and m for their values. It is required to find such values 
for them as will make the assumed development true for every 
possible value of z. 

If, in equation (1) we make ^ = 0, we have 

P= 1. 
Substituting this value for P, equation (1) becomes, 

{\-\-zY=\+ Qz -{- Bz^ + ^^3 4- &c. - - - (2). 
Equation (2) being true for all values of 2, let us make z:=y\ 
whence, 

(1 + 2/)- = 1 + ^y + Pz/2 + Sif + &c. - - - (3). 
Subtracting equation (3) from (2), member from member, and 
di\"iding the first member by (1 + ;2) — (1 -f y), and the second 
member by its equal z — ?/, we have, 

(i+)._(i + , ^ = ^^^ +^f!_:il! + st^ + &, . (4). 

(1+2) — (1+y) z-y z — y z — y 

If, now, we make 1 + 2 = 1 + y, whence z = y, the first 

member of equation (4), from previous principles, becomes 

m(l+0)"^^, and the quotients in the second member become 
respectively, 

/i^y\ ^ 1 /fl^n ^^,j{t^ :.3.^&C.&C. 

Substituting these results in equation (4) we have, 

w (1 + zy-^ ^q^lRz H- 3&2 + 47^23 _^ &c. - - - (5). 

Multiplying both members of equation (5) by (1 -\- z)^ we find, 



»i (1 -f z)"* = (2 + 2P 
+ Q 



+ 8>Sf 
+ 2P 



+ 4r 
+ 35 



+ &c. - - - (6). 



If we multiply both members of equation (2) by wi, we have 
m (1 + zY = m + mQz + mEz"^ + mSz^ + mTz"^ + &c. - - - (7). 

The second members of equatio:i^'^ (6) and (7) are equal to 
each other, since the first members are the same ; hence, we 
have the equation. 



m{-mQz-\-mE2^+mSz^-{-&c.-Q-^2E\z+SS\z^i-4T 

-i- ^1 +2Pl +3 5? 



23+&C- (8) 



260 ELEMENTS OF ALGEBRA. LCHAP. VIH 

This equation being identical, we have, (Art. 195), 



g = m, . 


- or. 


2i2+ ^ = w$,. 


or. 


3ASf+2i2 = mi2, 


- or, 


^T-\-ZS^mS, 


- or. 


&c., 


&;c., 



«=T 




m(m-l)_ 
■^-1.2' 




m(m — 1) (m 


-2). 


'^- 1.2. 


3 ' 



m{m—X){m — 2) (m — 3) 
~ 1.2.3.4 ^' 

&c. 

Substituting these values in equation (2), we obtain 

If now, in the last equation, we write — for z^ and then mul- 
tiply both members by a;'", we shall have, 

+ &c. . . (10). 

Hence, we conclude, since this formula is identical with that 
deduced in Art. 136, that the form of the development of (a; -[-«)'* 
will be the same, whether m is positive or negative, entire or 
fractional. 

It is plain that the number of terms of the development, when 
m is either fractional or negative, will be infinite. 

Applications of the Binomial Formula* 

204. If in the formula (x -f- a)*" == 

(a , m— 1 a^ , m —\ m — 2 a^ \ 



CKAP. VIII.] BINOmAL THEOREM 261 

1 1 I 

we make m = — , it becomes (x + «)" or ^-^ x -i- a — 

i_i _L 

-/, , 1 a . 1 n a^ , 1 n 

\ n X n Z x^ n 
or, reducing, 



■( 



1 a \ n — \ a^ \ 

1 H — . . — - — . -^ H — 

n X n Zii x^ 11 




The fifth term, within the parenthesis, can be found by mul- 
tiplying the fourth by — and by — , then changing the sign 

^Yh X 

of the result, and so on. 

205. The formula just deduced may be used to find an approx- 
imate root of a number. Let it be required to find, by means 
of it, the cube root of 31. 

The greatest perfect cube in 31 is 27. Let x = 27 and a = 4 : 
making these substitutions in the formula, and putting 3 in the 
place of n^ it becomes 

„ rrr^ .A . 1 4 1 1 16 . 1 1 5 64 



19683 



3 /3r= 3 H \ — 1- &c. 

V ^27 2187 ^ 531441 43046721 ^ 

Whence, ^T^ = 3 . 14138, w^hich, as we shall show presently, 
is exact to within less than .00001. 

We may, in like manner, treat all similar cases : hence, for 
extracting any root, approximatively, by the binomial formula, 
we have the* followinjr \ ^ 



RULE. 



\ 



Find the perfect power of the degree indicated, which is nearest 
to the given number, and place this in the formula for x. Sub- 
tract this power from the given number, and substitute this differ- 
ence, which will often be negative, in the formula for a. Perform 
the operations indicated^ and the result will be the required root. 



262 ELEMENTS OF ALGEBRA. [CHAP. VTEI. 



EXAMPLES. 



28 = 27*/l 4- ^) = 3.036G. 




2. %/W= (32 - 2)* = 32' A - ^) = 1.9744. 

3. ^/39"= (32 + if = 32''/l + ^V =: 2.0S07. 

1 

4. l/l08"= (128 - 20)^= 128^/l - 4) = 1.95204. 

206. When the terms of a series go on decreasing in value, 
the series is called a decreasing series ; and when they go on 
increasing in value, it is called an increasing series. 

A converging series is one in which the greater the number 
of terms taken, the nearer will their sum approximate to a 
fixed value, which is the true sum of the series. When the 
terms of a decreasing and converging series are alternately 
•positive and negative^ as in the fir;c example above, Tve can 
determine the degree of approximation when we take the sum 
of a limited number of terms for the true sum of the series. 

For, let a — b -\- c — d -\- e — /+ • • ♦? &c., be a decreasing 

series, h^ c^ d^ . . . being positive quantities, and let x denote. 

the true sum of this series. Then, if n denote the number of 

terms taken, the value of x will be found between the sums 

, of n and n -\- \ terms. 

For, take any two consecutive sums, 

c — h-{-c — d-\-e—f and a — b -\~ c — d {■ e — f 4- g. 

In the first, the terms which follow — /, are 
+ g-h, +/,_/-}-..; 
but, since the series is decreasing, the terms g — h^ k — I . , 
&c., are positive ; therefore, in order to obtain the complete 
value of x^ a positive number must be added to the sum 
a — h \- c — d -\- e — f. Hence, we have 

a — b + c — d-{-e — f<^x. 



CHAP. VlII.l BINOMIAL FORMULA. 263 

lu the second sum, the terms which follow -f- 9-, ' are - h 
-{- k — I -\- m . ... Now, — A + A:, — I -\- m . . . &c., are 
negative ; therefore, in order to obtain the sum of the series, 
a negative quantity must be added to ' , 

a — b + c — d + e - f -\- g \ 

or, in other words, it is necessary to diminish it. Consequently, 

a — b-\-c — d-{-e—f-\-gyx. 

Therefore, x is comiweliended between the sums of the first 
n and the first n -\- 1 terms. *' 

But the difference between these two sums is equal to g-, and 
isince X is comprised between them, g must be greater than 
the difference between x and either of them ; hence, the error 
committed by taking the sum of n ter.ms, a — b -\- c — d -\- e — /, 
of the series, for the sum of the series is numerically less than 
the following term. 

207. The binomial formula serves also to develop algebraic 
expressions into series. 

EXAMPLES, 

1. To develop the expression , we have, 

J. ■"" z 

[n the binomial formula, make m= — 1, a; = 1, and a =i; — Ik; 

and it becomes 

(1 _ ,)-! = 1 ^ 1 .(_,)_ 1 . ZllZli . (_,)2 
_1 -1-1 ^l^Hl (_,)3_ 

or, performing the operations indicated, we find for the de- 
velopment, 

Y~ = (1 - ^)"^ = 1 + a; + ^2 ^ 23 + ^ 4- &c. 

We might have obtained this result, by applying the rule 
for division. 



264 ELEMENTS OF ALGEBRA. CHAP. VUl 



2. Again take the expression, 
2 



or 2(1 -2)-3. 



(l-.)3 

. Substituting in the binomial formula — 3 for w, 1 fc r a:, 
and — z for a, it becomes, 

(i,-.-)-3 = i-3.(-^)-3.^:^.(-^)» 

3-1-3-2, ,, . 
- 3.— 2— —3— . {-i? - &e. 

Performing the indicated operations and multiplying by 2, 
we find 

.— ?— = 2 (1 + 32 + 6^2 + 10^3 + 150* + &c.). 

3. To develop the expression \J ^z — '^ we first place it 

under the form i/^22 x(l w]' ^1 '^^ application of the 

bmomial formula, we find 

~ 6 ^ ~ 36 ^ "" 648 ^ " • • • ' 

bence, ^2s - z^ = V^(^ ~ T ^ ~ 36 ^^ ~ 648 ^' ~' "^^7 

4. Develop the expression — = (a + ^)~^ hito a series 

5. Develop into a series. 

r -^ X 

/v2 /v3 /j^ 

^715. r — X -\ H T, &c. 

cfl -j. .-^2 

6. Develop the square root of — '— into a series. 

OC 3^ OC 

^2 

7. Develop the cube root of 7— -— into a series. 

(a^ 4- x^Y 



, 1 /, 22-2 , 5.r* 40.re ^ \ 



as 



CHAP. YIII.J SUMMATION OF SERIES. 265 

Summation of Series. 

208. The Summation of a Series, is the operation of finding 
an expression for the sum of any number of terms. Many 
useful series may be summed by the aid of two auxiliary series. 

Let there be a (/iven series, whose terms may be derived from 
the expression — — -- — -, by giving to p a fixed value, and then 
attributing suitable values to q and n. 

Let there be two auxiliary series formed from the expressions 

— and — ; — , so that the values of p, q, and n, fhall be the 
n ft -{- p 

same as in the corresponding terms of the first series. 

It can easily be shown that any term of the first series is 

e^ual to — multiplied by the excess of the corresponding term 

in the second series, over that in the third. 
For. if we take the expression 

p\n ?^ + p/^ 
and perform the operations indicated, we shall get the expression,^ 

hence, we have 

n{n -h p) p\n n -{■ p>/ ^ 
which was to be proved. 

It follows, therefore, that the sum of any number of terms oj 
tJte first series, is equal to — multiplied by the excess of the sum 

of the corresponding terms in the second series, over that of the 
corresp)onding terms in the third series. 

Whenever, therefore, we can find this last diflerence, it is 
always possible to sum the given series. 



I ~1 ■ ■ 



1 



266 ELEMENTS OF ALGEBRA. [CHAP. VIIL 



EXAMPLES. 

1. Requirec the sum of n terms of the series 

1.2^ 2.3 ^ 3.4 ^4.5^ 

Comparing the terms of this series with the expression 

9 

we see that making p = 1, <? = 1, and n = 1, 2, 3, 4, &c., in 
succession, will produce the given series. 

The two correspondmg auxiliary series, to n terms, are 



2 34 n ' n+1 

The difference between the sums of n terms of the first and 
second auxiliary series is 

1 7, or, if we denote the sum 

?i + 1 

of n terms of the given series by S, we have,- 

s=i- ' 



n + I 
If the number ^f terms is infinite n = co and 

S = 1. 
2. Required the sum of n terms of the series 

n + 375 + 577 + ^9 + 97ri + *'''•' 

If we compare the terms of this series with the expression 

9 
n(n 4- pY 

vre see that p =: 2, q = 1, and n = 1, 3, 5, 7, &c., in suc- 
cession. 



V^tjfV) 




/' 




■U^tU 


CHAP. VIII.J SUMMATION 


OF 


SERIES. 




267 ^- 


The two auxiliary series, to n 


ten 


ns, are, 

, 1 




Q 


• • ' 2» - 1' 


i-.i . 


•»" F+r + f+" 




•+2.-1^ 


1 




2w T 


1^ 



hence, as before. 

If w. = OD, we find S == — . 
/it 

3. Required the sum of n terms of the series 

1.4^2.5^3.6^4.7^ 
Here i? = 3, g' == 1, n = 1, 2, 3, 4, &;c. 

The two auxiliary series, to n terms, are, 

1,1 ,11,1,1. 

4"^ 5 '^ n~^ n-\- \'^n-\-2^ ?i+3' 



hence 



c_i/i , 1 , l__i 1 L_\ 

3 \ "^ 2 "^ 3 71 + 1 w + 2 ?i -h 3/' 

T^ e 11 

4. Required the sum of the series 

4 4 4 4 4 

fTs "^ 579 "^ 9.13 "^ 13. 17 "^ 17.21 "^" "^^^ 



V 



Ans» 1, 



5. Find the sum of n terms of the series, 

JL _ . A ^ ^ _ _5_ . _A_ _ & e 
3.5 5.7^7.9 9.11^11.13 ^^- * * 

Here p = 2, q =2, - 3, +4, - 5, -}- 6, &c 

»=: 3, 5, 7, 9, 11, &c. 



268 ELEMENTS OF ALGEBRA. LCflAP. VUI. 

The two auxiliary series are, 

_2_ _ _3_ _4 __5_ 71 -hi 



3 5'7 9' '2/i + l 

2 ^ , 4 , ^i _ 7t + 1 



'5 7 ' 9 271 -h 1 2/1 -i- 3 ' 

1/2 7i4-l\ 1/. ,., 

hence, 5 — ^( — ± 1 (1 — 1 + 1 — ... ±1). 

' 2 V 3 2;^ + 3/ 2 '^ ^ . . . =c i;. 

If 7i is even, the upper sign is used, and the quantity in 
the last parenthesis becomes + 1, in which case 

1 /2 ^+1 \ 1 _ 1 / 1, ri-\-\ \ 
~ 2 V 3 27^ + 3/ 2 "" 2 V 3 "^ 2/i + 3/* 

^ If /I is odd, the lower sign is used, and the quantity in tk 
last parenthesis becomes 0, in which case 

<? _ JL /I. _ Jl±l\ 
2 V 3 2/1 + 3/' 

If in either formula we make 

,, + 1 1+"^ , 1 . c 1 

^^ = ^'2^^^ = 3- ^^^^"^^^ ¥' ^^^ '^ = 12- 

6. Find the sum of n terms of the series, 

1.3 2.4~*'3.5 4.6' ^* 
Here, ^ = 2, g = 1, - 1, +1, - 1,+ 1, - 1, &;c. 

w = 1, 2, 3, 4, &c. 
■Rie two auxiliary series are, 

2 3 4 o 6 71, 

1.1 1^1 1.1. 1.^1 ' 

_3 4^5 6' ?i 7i + l n-fi 



whence, >S' = — ( — ^f — — - ±- — -— ). 

2 \ 2 n -\- \ n -}- 2/ 

If n = CO, we find S = -r. 

4 



CHAP. Till.] METHOD BY DIFFERENCES. 269 

Of the Method hy Differences, 

209. Let a, 6, c, fl? . . . . &c., represent the successive terms 
of a series formed according to any fixed law ; then if each 
term be subtracted from the succeeding one, the several re 
mainders will form a new series called the first order of dif- 
ferences. If we subtract each term of this series from the 
succeeding one, we shall form another series called the second 
order of differences^ and so on, as exhibited in the annexed 
table. 
a, 6, c, <?, e, 

6— a, c— &, <f— c, e — cf, &c., 1st. 

c-26+a, d-2c-\-h, e-2i+c,&c., 2d. 

c?-3c+36— a, e — 3cZ + 3c — ^>, &c., 3d. 

e — 4c? + 6c ~ 45 + a, &c., 4th. 

If, now, we designate the first terms of the first, second, 
third, &c. orders of differences, by d^^ c?2) ^s? ^4? <^c., we shall 
have, 

d^ — b— a, whence h = a-\- d^^ 

d.i = c — 2b -{- a, whence c = a + 2c?i + d^, 

cfg = c? — 3c + 36 — a, whence d =z a -i- Sd^ -{- Sd^ -{- d^^ 

c?4 = « — 4£^ + Cc — 46 + a, whence e = a + 4c?i + 6g?2 + ^^z + ^4, 
&c. &;c. &c. (fee. 

And if we designate the term of the series which has n 
terms bef:»re it, by T^ we shall find, by a continuation of 
the above process, 

+ 1.2.3.4 '^^ + *"'- - ■ - (!)• 

I 

This formula enables us to find the {n -\- \y^ term of a 
series when we know the first terms of the successive orders 
jf differences. 



270 ELEMENTS OF ALGEBRA. [CHAP. VJTI. 

210. To find an expression for the sum of n terms of the 
series a, 6, c, &c., let us take the series 

0, a, a + 5, a + 6 + c, a + 6 + c + c?, &c. - - - . (2) 
The first order of differences is evidently 

a, 6, c, c/, &c. (3) 

Now, it is obvious that the sum of n terms of the series (3), 
is equal to the [n-^l)*^ term of the series (2). 

But the first term of the first order of differences in series (2) 
is a ; the first term of the second order of differences is the 
same as c?,»in equation (1). The first term of the third order 
of differences is equal to d^^ and so on. 

Hence, making these changes in formula (1), and denoting the 
sum of n terms by S^ we have, 

S_na^-^^-^d,^ 1.2.3 ^^+ 1727374 ^^ 

-• &c. - - - - (4). 

When all of the terms of any orde^ of differences become 
equal, the terms of all succeeding ord^^rs of differences are 0, 
and formulas (1) and (4) give exact reroilts. When there are 
no orders of differences, whose terms become equal, then for- 
mulas do not give exact results, but approximations more or less 
exact according to the number of terms used. 

EXAMPLES. 

1. Find the sum of n terms of the series 1.2, 2.3, 3.4, 
4 . 5, &c. 

Series, 1.2, 2.3, 3.4, 4.5. 5 . 6, &c 

1st order of differences, 4, 6, 8, 10, &;c. 

2d order of differences, 2, 2, 2, &;c. 

3d order of differences, 0, 0. 

Hence, we have, a = 2, c?a — 4, d^ = 2, d^, d^, &c., equal 
lo 0. 



CHAP. VIII.. METHOD BY IIFFERENCES. 271 

Substituting these values for a, c7i, c?o, (fee, In formula (4), 

we find, 

^ ^ n(n — 1) , n(n — l)(n —2) 
S^2ni- ^ ^^^^^ X4+-i ^-^^-^ -^ X2; 

w .^ence, S = ^ !,^ . 

o 

2. Find the sum of n terms of the series 1.2.3, 2.3.4, 
3.4.5, 4.5.6, &c. 

1st order of differences, 18, 36, 60, 90, 126, &c. 

2d order of differences, 18, 24, 30, 36, &c. 

3d order of differences, 6, 6, 6, &c. 

4th order of differences, 0, 0. &c. 

We find a = Q, d, = 18, d, = 18, ^3=6, d, = 0, &;c. 

Substituting in equation (4), and reducing, we find, 

_, n{n H- 1) (71 + 2) {n + 3) 
^ — . 

4 ! 

3. Find the sum of n terms of the series 1, 1+2, l+2-(-3, 
1 -f- 2 + 3 + 4, (fee. 

Series, 1, 3,. 6, 10, 15, 21. 

1st order of differences, 2, 3, 4, 5, 6. 

2d order of differences, 1, 1, 1, 1. 

3d order of differences, 0, 0, 0, 

a = 1, c?i = 2, d^ = 1, fl?3 = 0, d^ = 0, &c. ; 

n(n—l) ^ n(n-])(n—2) n^ + Sn"^ + 2n 
hence, S =n ^ ±-^,2 + ^^^ ^ ^ ^^ ; 

T . e r.(7i+ l)(w + 2) 
or, reducmg, aS = — j— ^-7^ 

4. Find the sum of n terms of the series P, 2^, 3^, 4."^, 5-, &c. 

We find, a = 1, <;, = 3, tf, = 2, c/^ = 0, d^ = 0, &c., &c. 

Substituting these values in formula (4), and reducing, we find, 

_ n{n+ l)(2/2 + 1) 
1.2 3 



272 



ELEMENTS OF ALGEBRA. 



[CHAP. vm. 



5. Find the sum of n terms of the series, 

1 . (m + 1), 2 {in + 2), 3 [m + 3), 4 {m + 4), &c. 
We find, a = ??i + 1, c?i = m + 3, c?2 = 2, cZj r= 0, &c. ; 



whence. S ^n (m + 1) 4 



n.{n — l) 



1 



;m + 3) + 



n.(7i — l)(n— 2) 



1.2.3 



X2, 



or. 



5 = 



n.(?i + l).(l+2ri+3m) 
1.2.3 

Of Piling Balls. 



^T 



The last three formulas deduced, are of practical appli- 
cation in determining the number of balls in different shaped 
piles. 

First^ in the Triangular Pile. 

211. A triangular pile is formed of succces- 
sive triangular layers, such that the number 
of shot in each side of the layer?, decreases 
continuously by 1 to the single shot at the 
top. The number of balls in a complete tri- 
angular pile is evidently equal to the sum 
of the series 1, 1 + 2, 1 + 2 + 3, 1+2 + 3 
+ 4, &c. to 1 + 2 + . . . + w, fi denoting the number of balls 
on one side of the base. 

But from example 3d, last article, we find the sum of n 
terms of the series. 




n{n-\-\ ) {n + 2) 
'^^1.2.3 

Second^ in the Square Pile. 
212. The square pile is formed, 
63 shown in the figure. The num- 
ber of balls in the top layer is 1 ; 
the number in the second layer is 
denoted by 2^ ; in the next, by 3^, 
and so on. Hence, the number of 
balls in a pile of n layers, is equal 
to the sum of the series, P, 2^ 3^, 



(!)■ 




9« ft® #• •# • 



CHAP. VIII.] PILING BALLS. 273 

Vc, n^, which we see, from example 4th of the last article, is 

■ • - (2)- 

Third, in the . Oblong Pile. 



».(>. + l).(2« + l) 
^= 1.2.3 




213. The complete oblong pile has {m + 1) balls in the 
upper layer, 2.(m -\-2) in the next layer, 3 (w + 3) in the 
third, and so on : hence, the number of balls in the complete 
pile, is given by the formula deduced in example 5th of the 
preceding article, 

n . { n-i- 1).(1 +2yi -\-Sm) 

1 ; 2 ; 3 



S=- 



(3). 



214. If any of these piles is incomplete, compute the num- 
ber of balls that it would contain if complete, and the number 
that would be required to complete it ; the excess of the fop- 
mer over the latter, will be the number of balls in the pile. 

The formulas (1), (2) and (3) may be written. 



triangular, S =: ^ . !!:l!!:±}l (n + 1 .^ 1) 

square, S = ~. —^ (t? + ?i -f 1) 

o /4 



(1); 
(2); 



rectangular, S = 



Now, since 



1 ^^(^+ 1) 
3"' 2 



((n-+-m) + (n+m)-f (m+l)V (3). 

n (n -f 1) . , , /. -, 1, . 1 

— ^—- IS the number of balls m the tn- 

<^ 

angular face of each pile, and the next factor, the number of balls 

in the longest line of the base, plus the number in the side 

of the base opposite, plus the parallel top row, we have th« 

following 

18 



\ 



274 * ELEMENTS OF ALGEBRA. [CHAP. Vin. 



RULE. 

Add to the number of balls in the longest line of the base t?i« 
number in the parallel side opposite, and also the number in the 
top parallel row ; then 7nultiply* this sum by one-third the number 
in the triangular face ; the product will be 'the humber of halls in 
the pile. 

EXAMPLES. (y> 

1. How many balls in a triangular pile of 15 courses 1 

Ans. 680." 



2. How many balls in a square pile of 14 courses ? and how 
many will remain after 5 courses are removed 1 

Ans. 1015 and 960. 

3. In an oblong pile, the length and breadth at bottom are 
respectively 60 and 30 : how many balls does it contain ? 

Ans. 23405. 

4. In an incomplete oblong pile, the length and breadth 
\ / at bottom are respectively 46 and 20, and the length and 

breadth at top 35 and 9 : how many balls does it contain 1 

. Ans. 7190. 

"""^ . .^ 

5. How many balls in an incomplete triangular pile, the num 

ber of balls in each side of the lower course being 20, and 
iu each side of the upper, 10 J? 

6. How many balls in an incomplete square pile, the number 
iu each side of the lower course being 15, and in each side 
of the upper course 6 ? 

7. How many balls in an incomplete oblong pile, the num- 
bers in the lower courses being 92 and 40 ; and the numbers 
in the :orresponding top courses being 70 and 18 1 







CHAPIER IX. 

OOirriNUED FRACTIONS — EXPONENTIAL QUANTITIES LOGARirHMS, IND 

FORMULAS FOR INTEREST. 

215. Every expression of the form 

j J 1 

a+l' a+l' a+1' 



h 6+1 6+1 



c c+1 

d 

in which a, 6, c, e?, &c., are positive whole nuirbers, is called a 

continued fraction : hence, 

A CONTINUED FRACTION ktts 1 foT its numerator, and for its de- 
nominator, a whole number plus a fraction, which has 1 for its 
numerator and for its denominator a whole number plus a fratx- 
iijn, and so on. 

216. The resolution of equations of the form 

a* = b, 

gives rise to continued fractions. 

Suppose, for example, a = 8, 5 = 32. We then have 

8* = 32, 
ill which ar > 1 and less than 2. Make 

. = 1 + 1. 



276 ELEMENTS OF ALGEBRA. LCHAP. tS. 

ill which 21 > 1, and the proposed equation becomes 

i+i i 

32 = 8 y=8x8y; whence, 

i y 

8^ = 4, and consequently, 8 = 4. 

It is plain, that the value of y lies between 1 and 2. Suppose 





y=i + -J-, 


and we have, 


8 = 4 '=4x4'; 


Lence, 


4» = 2, and 4 = 2*, or z = 2. 


But, 


y = '^\ = ^^\A-' 


and 


^_l+l_14. 1 -1+2 _5. 


^-1+3,-1+ 1-1+ 3-3. 
■•"2 


and this value will satisfy the proposed equation. 




A . r^ . r.-^ . r.^^ A „ 



For, 8^ = 3/85 = ^/pp = 3/(2^)3 =: 2 =32. 

21 7« If we apply a similar process to the equation 
(10)' = 200, 
we shall f^nd 

a; = 2-l-l; 2^ = 3-f-; ^=3 + -. 
y z u 

Since 200 is not an exact power, x cannot be exactly ex- 
pressed either by a whole number or a fraction: hence, the 
value of X will be incommensurable with 1, and the continued 
fraction will not terminate, but will be of the form 

ar = 2 + -=2 + ^ ^ = 2 + ^ 



3 + -i- 3-f ^ 



u 4- &c. 



OHAP, IX. 1 CONTINUED FRACTIONS. 277 

21 8i Vulgar fractions may also be placed un3er the form of 

continued fractions. 

65 
Let us take, for example, the fraction — ^, and divide both 

its terms by the numerator 05, the value of the fraction will 
not be cnanged, and we shall have 

65 _ 1 

149 ~ 21?' 
65 

^ ...... 65 1 

or enectmff the division, — — = 77-. 

^ 149 19 

^ + 65 

19 

Now, if we neglect the fractional part, — , of the denomina 

toi, we shall obtain — for an approximate value of the given 
tit 

fraction. But this value will be too large, since the denomina- 
tor used is too small. 

x9 

If, on the contrary, instead of neglecting the part — , we 

00 

were to replace it by 1, the approximate value would be -—, 

o 

which would be too small, since the denominator 3 is too 

large. Hence, 

1 65 ^ 1 ^ 65 

■2->l49 ^^^ ■3-<l49' 

therefore the value of the fraction is comprised between — and — -. 

2> 3 

If we wish a nearer approximation, it is only necessary to 

19 65 

operate on the fraction — r as we did on the given fraction -r-r?:, 
6d ° 149 

aJid we obtain, 

19 1 

^^~3+-^. 
^+I9' 



hence, 



65 
149 



' + w 



Jig ELEMENTS OF ALGEBRA. 'CHAP. IX. 

o 
If, now, we neglect .ne part — , the denoniinator 3 will be less 

thaD the true denominator, and — will he larger than the num 

o 

ber which ought to be added to 2 ; hence, 1 divided by 2 -h — 

will be less than the true value of the fraction ; that is, if we 
stop after the first reduction and omit the last fraction, the 
result will be too great ; if at the second, it will be too small, (fee. ; 
and, generally, 

If we stop at an odd reduction, and neglect the fractional part 
that comes after, the result will be too great; hut if we stop at 
an even reduction, and neglect the fractional part that follows, the 
result will he too small. 

219. The separate fractions — , -r-, — , (fee, which make up 

a c 

a continued fraction, are called integral fractions. 
The fractions. 



1 


1 


^ ,,.c. 


a"' 


1 ' 


c* 


«+4 


c 





are called approximating fractions, because each gives, in succes- 
sion, a nearer approximation to the true value of the fraction : 
honce, 

An approximating fraction is the result obtained by stopping 
at any integral fraction, and neglecting all that come after. 

If we stop at the first integral fraction, the resulting approxi- 
mating fraction is said to be of the first order ; if at the second 
integral fraction, the resulting approximating fraction is of the 
second order, and so on. 

When there is a finite number of integral fractions, we shall 
get the true value of the expression by considering them all : 
when their number is infinite, only an approximate value can be 
found. 



CHAP. IX.] CONTINUED FRACTIONS. 279 

220. We will now explain the manner in which any approxi- 
niatmg fraction may be found from those which precede it. 

(3). 



^6 



I 

1 



a 


1st app. fraction. 


b 
ab-{- 1 


2d app. fraction. 


bc+l 


— 3d app. fractior.. 



a + 



^4- 



1 
c 

By examining the third approximating fraction, we see that 
Its numerator is formed by multiplying the numerator of tho 
preceding approximating fraction by the denominator of the 
third integral fraction, and adding to the product the numerator 
of the first apjDroximating fraction : and that the denominator 
is formed by multiplying the denominator of the preceding 
approximating fraction by the denominator of the third integral 
fraction, and adding to the product the denominator of the 
first approximating fraction. 

Let us now assume that the {n — 1)*'^ approximating fraction 
is formed from the two preceding approximating fractions by the 
P Q . R 

(n — 3), {n — 2), and {n — 1), approximating fractions. 

Then, if m denote the denominator of the {n — 1)** integral 
fraction, we shall have from the assumed law of formation, 
R Qrn+P 



same law, and let — , — , and — , designate, respectively, the 



R ~ Q'm +P" 



(1). 



1 S^ 

Let us now consider another integral fraction — , and suppose 7- 

n Sr 

to represent the n^^ approximating fraction. It is plain that 

o 7? 

we shall obtain the value of — , from that of — , bv <^imply 

changing — ijito —, or, by substituting mH — for m, in 

m \ 

n 

♦equation (1); 



280 



ELEMENTS OF ALGEBRA. 



[CHAP. IX. 



whence, — = 
o 



(^'^i) 



-hP 



c>f + ^) + i^^ 









Hence, if the law assumed fcr the formation of the (w — 1)** ap- 
proximating fraction is true, the same law is true for the forma- 
tion of the n*^ approximating fraction. But we have shown 
that the law is true for the formation of the third; hence, it 
must be true for the formation of the fourth, being true for 
tlK^ fourth, it is true for the fifth, and so on ; neuce, it is gen 
tral. Therefore, 

The numerator of the n*^ approximating fraction is formed by 
multiplying the numerator of the preceding fraction by the denom 
inator of the n*^ integral fraction, and adding to the product the 
numerator of the [n — 2)^'' approximating fraction ; and the denom- 
inator is formed according to the same law, from the two preceding 
denojninators. xJ 

221. If we take the difference between the first and second 
approximating fractions, we find, 

1 b __ob-{-l —ab _ +1 

T ~ a5 + 1 ~ a{ab + 1) "~ a(ab + 1) ' 

aiid the difference between the second and third is, 

b bc-^1 - 1 



ab + 1 {ab -f Ijc -f a {ab + 1) [{ab -\- l)c+ a]' 
In both these cases we see that the difference between two 
consecutive approximating fractions is numerically equal to 1, 
divided by the product of the denominators of the two fractions 
To show that this law is general, let 

^ Q_ ^ 
p^ ^/» ^/. 

be any three consecutive approximating fractions. TheD 
P _ £ _ PQ'-P'Q ^ 

Q_ _R _ R'Q—RQ' 



aiid 



CHAP. IX. J CONTINUED FRACTIONS. 28] 

But Ii= Qm + F, and R' = Q'm -{- P' (Art. 220). 
Substituting these values in the .tist equation, we have, 

q' R' R'Q' 

or, reducing, 

q' R- RQ' ' 
Now, if (FQ' - F'Q) is equal to dz 1, then {F'Q — FQ') must 
be equal to =F 1 ; that is. 

If the difference between the {n — 2) and the [n — 1) fractions^ 
is formed hj the assumed laiv, then the difference between the 
{n — 1)*'' and the n*^ fractions must be formed by the same law. 

But we have shown that the law holds true for the difference 
between the second and third fractions ; hence, it must be true for 
the difference between the third and fourth; being true for the 
difference between the third and fourth, it must be true for the 
difference between the fourth and fifth, and so on ; hence, it is 
general : that is. 

The difference between any two consecutive approximating frac- 
tions, in- equal to dz 1, divided by the product of their denom- 
inators. 

When an approximating fraction of an even order v^ taken 
from one of an odd order, the upper sign is used : when one 
of an odd order is taken from one of an even order, Lhe> 
lower sign is used. 

This ought to be the case, since we have shown that ever^ 
approximating fraction of an odd order is greater than the true 
value of the continued fraction, whilst every one of an even 
order is less. 

222. It has already been shown (Art. 218), that each of the 
approximating fractions of an odd order, exceeds the true value 
of the continued fraction ; while each one of an even order 
is less than it. Hence, the difference between any tvro con- 
secutive approximating fractions U greater than the difference 



282 



ELEMENTS OF ALGEBRA. 



LCHAP. IX. 



between either of tliem and the true value of the continued 
fraction. Therefore, stopping at the n^^ approximating fraction, 
the result will be the true value of the fraction, to within less 
than 1 divided b}' the denominator of that fraction, multiplied 
by the denominator of the approximating fraction which follows. 

Tims, if Q^ and R^ are the denominators of consecutive ap- 
proximating fractious, and we stop at the fraction whose de- 
nominator is Q', the result will be true to within less than « 

^ -" 

But, since a, b, c, c?, &c,, are entire numbers, the denominatoi H' 
will be greater than Q\ and we shall have 

— < — • 



hence, if the result be true to within less than 



'R" 



Lt will 



certainly be true to within less than the larger quantity 

1 1 . 

--^-r ; that IS, 

The approximate result which is obtained, is true to within 
less than 1 divided by the square of the denominator of the laat 
ai>proximating fraction that is employ/ ed. 

829 
223. If we take the fraction — ^, v/e have, 



829 
347 



2 + 



2 + 



1 + 



1+ 1 



'*h 



Hero, we have in the quotient the whole number 2, which 
may either be set aside, and added to the fractional part after 
its value shall have been found, or we may place 1 under it 
for a denominator, and fe»eat it as an approximating fraction. 



CHAP. IXJ EXPONENTIAL QUANTITIES. 283 

ScluHon of the Equation a' = 6. 

224. Ai equation of the form, 

a' = 6, 
^ called an exponential equation. The object in solving this 
equation i&, to find the exponent of the power to which it is 
necessary to raise a given number a, in order to produce 
another given number h. 

225. Suppose it were required to solve the equation, 

2 = 64. 
By raising 2 to its different powers, we find that 

6 

2 r= 64 ; hence, a; = 6 
will satisfy the equation. 

Again, let there be the equation, 

3 = 243, in which x=zh. 

Now, so long as the second member 5 is a perfect power of 
the given number a, the value of x may be obtained by trial, 
by raising a to its successive powers, commencing at the first, 
f)r the exponent of the power will be the value of x. 

226. Suppose it were required to solve the equation, 

z 

2 ==6. 
^Y making a; = 2, and ar = 3, we find, 

2 3 

2 =4 and 2=8; 
from which we perceive that the value of x is comprised be^ 
twee'i 2 and 3. 

Make, then, x =2 -] -^ in which ir^ > 1. 

Substituting this value in the given equation, it becomes, 

2+1 ± 

2 ^' ~ 6, or 22 X 2^^ = 6 ; hence, 

2^--^ -A. 
- 4 ~ 2 ' 



284: ELEMENTS OF ALGEBRA. [CHAP. IX. 

and by changing the order of the members, and raising both 
to the a/ power, 

To determine a/, make 3/ successively equal to 1 ari 2; we 
find, 

therefore, a/ is comprised between 1 and 2. 
Make, a:' = 1 -|- -7;, hi which a/^ > 1. 

By substitutmg this value in the equation I — j =2, 

^e find, (!)'+- = 2 ; hence, |- x g-)^ = 2, 

and consequently, I — J =— . 

4 3 

The supposition, a/^ = l, gives -7r<^7r5 

-1 /» • > 

and a/' = 2, gives -^ > -^ y 
therefore, 3/^ is comprised between 1 and 2. 
Let x'' z=l-] — ; then, 

, /9\^'" 4 

whence, I — | = —. 

\S) 3 



If we make a/^^ = 2, we have 

4 
3 



< o , 



\s) ~ 64 

and if we make a/''^ — 3, we have 

/9\3_729 4 ^ 
\8/ ""512-^ 3 * 



CHAP. IX. J EXPONENTIAL QUANTITIES. - 285 

therefore, x'^^ is comprised between 2 and 3. 
Make ocf'^ = 2 H -^ and we have 

(-) - ^-; hence, ^(-3-).^^^^; 

and consequently, (9z^) ~ "ft"* 

Operating upon this exponential equation in the same manner 
as upon the preceding equations, we shall find two entire num 
bers, 2 and 3, between which a;^^ will be comprised. 

Making 

j;lV ^ 2 + -1, 

X can be determined in the same manner as a:^^, and so on. 
Making the necessary substitutions in the equations 

r:=2+4;, ^=l+-77» ^^' = 1 + 37/. ^'"=2 + -^^ . . . ., 
xf • x" xf" x^"^ 

we obtain the value of x under the form of a whole number, 
plus a continued fraction. 

1 



a; = 2 + 



-^^. 



2 + 1 



hence, we find the first three approximating fractions to be 

J_ JL A 

1' 2' 5' 

and the fourth is equal to 

3x2+1 _ 7 

5x2 + 2 - 12 ^^''- ^^°)' 

which is tne true value of the fractional part of a? to within 
less than 



286 ELEMENTS OF ALGEBRA, [CHAP. IX 

Therefore, 

7 31 1 

x = 2 + —=— = 2.58333 + to within less than — -, 
12 12 144 

and if a greater degree of exactness is required, \re must take 
a greater number of integral fracticns. 

EXAMPLES, 

3 = 15 - - a; = 2.46 to within less than 0.01. 



(10) =3 . • - x= 0.477 " " 0.001, 

2^ 

3 



b' = ~ ' ' ' x= - 0.25 " " 0.01. 



Of Logarithms. 

227. If '^e suppose a to preserve a constant value in the 

equation 

a' = iV, 

whilst iV is made, in succession, equal to every possible num- 
ber, it is plain that x will undergo changes corresponding to 
those made in JV. By the method explained in the last arti- 
cle, we can determine, for each value of iV", the corresponding 
value of X, either exactly or approximatively. 

The value of x, corresponding to any assumed value of the 
number iV, is called the logarithm of that number ; and a is 
called the base of the system in which the logarithm is taken. 
Hence, 

The logarithm of a numher is the exponent of the power to which 
it is necessary to raise the base, in order to produce the given number. 
The logarithms of all numbers corresponding to a given base constitute 
a system of logarithms. 

Any positive number except 1 may be taken as the base 
Df a system of logarithms, and if for that particular base, we 
suppose the logarithms of all numbers to be computed, they 
V7ill constitute what is called a system of logarithms. Hence, 
we see that there is an infinite n:mber of systems of loga- 
rithms. 



CHAP. IX.] THEORY OF LOGAKIl HMS. 287 

228. The base of the common system of logarithms is 10, 
and if we designate the logarithm of any number taken in 
itint system by log, we shall have, 



(10)0 = 


1 


whence, 


l02 1=0 


(10)1 ^ 


10 


whence, 


log 10 = 1 


(10)2 ^ 


100 


whence, 


log 100 = 2 


(10)3 = 


1000; 


whence. 


log 1000 = 3 



(fee, &c. 

We see, that in the common system, the logarithm of any 
number between 1 and 10, is found between and 1. The 
logarithm of any number between 10 and 100, is between 1 and 
2 ; the logarithm of any number between 100 and 1000, is be- 
tween 2 and 3 ; and so on. 

The logarithm of any number, which is not a perfect power 
of the base, will be equal to a whole number, plus a fraction, 
the value of which is generally expressed decimally. The entire 
part is called the characteristic^ and sometimes the index. 

By examining the several powers of 10, we see, that if a 
number is expressed by a single figure, the characteristic of its 
logarithm will be ; if it is expressed by two figures, the 
characteristic of its logarithm will be 1 ; if it is expressed by 
three figures, the characteristic will be 2 ; and if it is expressed 
by n places of figures, the characteristic will be n — \. 

If the number is less than 1, its logarithm will be negative, 
and by considering the powers of 10, which are denoted by 
negative exponents, we shall have, 

(10) = — = -1 7 whence, log .1 = — 1. 

-2 1 

(10) = = .01 ; whence, log .01 == ~ 2. 

-3 1 

(10) = = .001 ; whence, loor .001 = — 3. 

^ ^ 1000 '6 

&c., (kc. &c., (fee. 

Here we see that the logarithm of every number between 1 and 

.1 will be found between and — 1 ; that is, it will je equal to 

— 1, plus a fraction less than 1. The logarithm of any number 



288 



ELEMENTS OF ALGEBRA. 



[CHAP. IX. 



between .1 and .01 will be between —1 and —2; that is, it 
will be equal to — 2, plus a fraction. The logarithm of any 
number between .01 and .001, will be between — 2 and — 3, 
or will be equal to — 3, plus a fraction, and so on. 

In the first case, the characteristic is — 1, in the second — 2, 
in the third — 3, and in general, the characteristic of the logarithm 
of a decimal fraction is negative^ and numerically 1 greater than 
the number of O's which immediately follow the decimal point. The 
decimal part is always positive, and to indicate that the negative 
sign extends only to the characteristic, it is generally written 
over it; thus, 
log 0.012 = 2.079181, which is equivalent to — 2 + .079181. 

228*» A table of logarithms, is a table containing a set of 
numbers, and their logarithms so arranged that we may, by its 
aid, find the logarithm" of any number from 1 to a given num- 
ber, generally 10,000. 

The following table shows the logarithms of the numbers, from 
1 to 100. 



N. 


Los. 


N. 


Lng. 


N. 


Log. 


i\. 


Log. 


1 


0.000000 


26 


1.414973 


51 


1.707570 


76 


1.880814 


2 


0.301030 


27 


1.431364 


52 


1.716003 


77 


1.886491 


3 


477121 


28 


1.447158 


53 


1.724276 


78 


1.892095 


4 


0.602060 


29 


1.462398 


54 


1.732394 


79 


1.897627 


5 


0.698970 


30 


1.477121 


55 


1.740863 


80 


1.903090 


~6 


0.778151 


31 


1.491362 


56 


1.748188 


81 


T. 908485 


' n 


0.845098 


32 


1.505150 


57 


1.755875 


82 


1.913814 


8 


0.903090 


33 


1.518514 


58 


1.763428 


83 


1.919078 


9 


0.954243 


34 


1.531479 


59 


1.770852 


84 


1.924279 


10 


1.000000 


35 


1.544068 


60 


,1.778151 


85 


1.929419 


TT 


1.041393 


36 


1.556303 


61 


1.785330 


86 


1.934498 


12 


1.079181 


37 


1.568202 


62 


1.792392 


87 


1.939519 


13 


1.113943 


38 


1.579784 


63 


1.799341 


88 


1.944483 


14 


1.146128 


39 


1.591065 


64 


1.806180 


89 


1.949390 


15 


1.176091 


40 


1.602060 


65 


1.812913 


90 


1.954243 


16 


1.204120 


41 


1.612784 


66 


1819544 


91 


1.959041 


17 


1.230449 


42 


1.623249 


67 


1.826075 


92 


1.963788 


18 


1.255273 


43 


1.63^468 


68 


1,832509 


93 


1.968483 


19 


1.278754 


44 


1 613453 


69 


1838S49 


94 


1.973128 


20 


1.301030 


45 


1.653213 


70 


1.845098 


95 


1.977^24 


21 


1322219 


46 


1.662758 


71 


1851258 


96 


1,982271 


22 


1.342423 


47 


1.672098 


72 


1.857333 


97 


1986772 


23 


1.361728 


48 


1.681241 


73 


1.868323 


98 


1991226 


24 


1.380211 


49 


1.690196 


74 


1.869232 


99 


1.995635 


25 


1.397940 


50 


1.698970 


75 


1.875061 


100 


2.000000 



CHAP. IX.] THEORY OF LOGARITHMS." 289 

When the number exceeds 100, the characteristic of its loga- 
rithm is not written in the table, but is always known, since 
it is 1 less than the number of places of figures of the given 
number. Thus, in searching for the logarithm of 2970, in a table 
of logarithms, we should find opposite 2970, the decimal part 
.472756. But since the number is expressed by four figures, 
the characteristic of the logarithm is 3. Hence, 

log 2970 = 3.472756, 
Rnd by the definition of a logarithm, the equation 
a' =z N^ gives 
103.472; 60 -_ 2i^7o. 

General Properties of Logarithms. 

229. The general properties of logarithms are entirely inde- 
pendent of the value of the base of the system in which they 
are taken. In order to deduce these properties, let us resume 
the equation, 

in which we may suppose a to have any positive value ex- 
cept 1. 

230. If, now, we denote any two numbers by N' and i^',, 
and their logarithms, taken in the system whose base is a, 
by x' and a/', we shall have, from the definition of a logarithm, 

a'' z=zN' (1), 

and, a'"=N'' (2). 

If we multiply equations (1) and (2) together, member by- 
member, we get, 

ax'+x" =,N' X N'' - . - (3). 

But since a is the base of the system, we have from the 

definition, 

a/ + a/' = log {N' X N'') ; that is, 

The logarithm of the product of two numbers is equal to the 

sum of their logarithms. 

19 



290 ELEMENTS OF ALGEBRA. [CHAP. IX 

' 231 • If we divide equation (1) hj equation (2), member by 
lu ember, we have, 

«"-"' = IV^. (4) 

15 ut, from the definition. 

a/-a/^ = log^-^j; that is, 

The logarithm of the quotient which arises from dividing om 
number hy another is equal to the logarithm of the dividend viimts 
the logarithm of the divisor. 

232. If we raise both members of equation (1) to the n** 
power, we have, 

anz' _ iV"/" ..... (5). 

But from the definition, we have, 

nx' = log (iV''«) ; that is. 

The logarithm of any power of a number is equal to the 
logarithm of the number 7nultiplied by the exponent of the power. 

233. If we extract the n^^ root of both members of equation 
(1), we shall have, 

a« ={Ny= yW . - (6). 
But from the definition, 

a/ 

— = log {\/W) ; that is, 
n » 

The logarithm of any root of a number is equal to the loga- 
rithm of the number divided by the index of the root. 

234. From the principles demonstrated in the four preceding 
articles, we dedude the following practical rules : — 

Firsts To multiply quantities by means of their logarithms. 

Find from a table, the logarithms of the given factors, take 
the sum of these logarithms, and look in the table for the cor. 
responding number ; Ms will be the product required. 



CHAP. IX.J ' THEORY OF LOGARITHMS. 291 

Thus, log 7 0.845098 

log 8 0.003090 

log 56 1.748188 ; 

hence, 7 x 8 = 56. 

Second. To divide quantities by means of their logarithms. 

Find the logarithm of the dividend and the logarithm of the 
divisor, from a table ; subtract the latter from the former, and 
look for the number corresponding to this difference ; this will be 
iJie quotient required. 

Thus, log 84 - 1.924279 

log 21 1.322219 

log 4 0.602060 ; 

hence, — = 4. 

Third, To raise a number to any power. 

Find from a table the logarithm of the number, and multiply it 
by the exponent of the required power ; find the number correi- 
ponding to this product, and it will be the required power. 

Thus, log 4 0.602060 

3^ 

log 64 1.806180 ; 

hence, (4)^ = 64. 

Fourth, To extract any root of a number. 

Find from a table the logarithm of the number, and divide 
this by the index of the root ; fnd the number corresponding to 
this quotient, and it will be the root required. 

Thus, log 64 1.806180(6 

log 2 ....,- .301030; 

hence, ^/^64 = 2. 

By the aid of these principles, we may write the folio win jf 
equivalent expressions : — 



292 ELEMENTS OF ALGEBRA. [CHAP. IX. 

Log [a .b .€ . d . . . .) = log a + log b + log c . . . . 

Log l-^j = log a + log 6 + log c — log c? -- log «l 

Log (a^ .b^ .cP ) = m log a -\- n log b -\- p log c -f- . . . . 

Log (a2 — a:^) = log (a + a;) + log (a — x). 

Log y (a2 — x^) = ^ log (a + re) + J log {a — x). 

Log (a^xl/^) =3flog«. 

234, We have already explained the method of determining 
the characteristic of the logarithm of a decimal fraction, in the 
common system, and by the aid of the principle demonstrated 
m Art. 231, we can show 

That the decimal part of the logarithm is the same as the decimal 
part of the logarithm of the numerator^ regarded as a whole number. 

For, let a denote the numerator of the decimal fraction, and 
let m denote the number of decimal places in the fraction, then 
will the fraction be equal to 

a 

and its logarithm may be expressed as follows: 

log — — = log a — log (10)'" = log a — m log 10 = log a — m , 

but m is a whole number, hence the decimal part of the loga 
rithm of the given fraction is equal to the decimal part of 
log a, or of the logarithm of the numerator of the giveu 
fraction. 

Hence, to find the logarithm of a decimal fraction from the 
common table. 

Look for the logarithm of the number, neglecting the decimal 
point, and then prefix to the decimal part found a negative charac- 
teristic equal to 1 moi'e than the number of zeros which immediately^ 
follow the decimal point in the given decimal. 

The rules given for finding the characteristic of the logarithms 
taken in the common system, will not apply in any other 
system, nor could we find the logarithm of decimal fractions 



CHAP. IX.: THEORY OF LOGARITHMS. 293 

directly from the tables in any other system than that whose base 
is 10. 

These are some of the advantages which the common system 
possesses over every other system. 

235. Let us again resume the equation 

1st. If we make iV= 1, a; must be equal to 0, since a^ = 1 ; 
that is, 

The logarithm of 1 in any system is 0. 

2d. If we make iV=a, x must be equal to 1, since a^ =za' 
that is, 

Whatever be the base of a system^ its logarithm, taken in that 
system, is equal to 1. 

Let us, in the equation, 

First, suppose a > L 

Then, when ]Sf= 1, re = 0; when iV> 1, a: > ; when iV< 1, 
r < 0, or negative ; that is, 

In any system whose base is greater than 1, the logarithms of 
all numbers greater than 1 are positive, those of all numbers less 
than 1 are negative. 

If we consider the case in which iV"-< 1, we shall have 

a-* = iV. or — = N, 
' a' 

Now, if N diminishes, the corresponding values of x . must 
increase, and when N becomes less than any assignable quan- 
tity, or 0, the value of x must be oo : that is, 

The logarithm of 0, in a system whose base is greater than I, 
is equal to — oo. 

Second, suppose a < 1. 

Then, when iV= 1, a: = 0; when iV^< 1, a: > ; wheniV>l, 
a? < 0, or negative : that is, 



\ \.^^^ 



^^^sUiJU 



M 



294 ELEMENTS OF ALGEBRA. [CHAP. IX 

In any system whose base is less than 1, the logarithms of all 
numbers greater than 1 are negative, and those of all numbers less 
than 1 are positive. 

If we consider the case in which iV< 1, we shall have a* = iV, 
in which, if N be diminished, the value of x must be increased ; 
and finally, when iV == 0, we shall have x = co: that is. 

The logarithm of 0, in a system whose base is less than 1, it 
equal to -\- 00. 

Finally, whatever values we gi^e to x^ the value of a' or 
N will always be positive; whence we conclude that negative 
numbers have no logarithms. 

Logarithmic Series. 
236 • The method of resolving the equation, 

explained m Art. 226, gives an idea of the construction of loga- 
rithmic tables ; but this method is laborious when it is necessary 
to approximate very near the value of x. Analysts have dis- 
covered much more expeditious methods for constructing new 
tables, or for verifying those already calculated. These methods 
consist in the development of logarithms into series. 
If we take the equation, 

a' = 2/, 

and regard a as the base of a system of logarithms, we shall 

have, 

log y z=zx. 

The logarithm of y will depend upon the value of y, and 
also upon a, the base of the system in which the logarithms 
are taken. 

Let it be required to develop log y into a seiies arranged 
according to the ascending powers of y, with co-efficients that 
are independent of y and dependent upon a, the base of the 
system. 



I 



CHAP. IX.] LOGAEITHMIC SERIES. 295 

Let us first assume a development of the required form, 
log y = J/+ Ny + Py2 + ^^3 ^ &c., 

ill which if, iV, P, &c. are independent of y, and dependent 
upon a. It is now required to find such values for these co- 
efliicients as will make the development true for every value 
of y. 

Now, if we make y z= 0, log y becomes infinite, and is either 
negative or positive, according as the base a is greater or less 
than 1, (Arts. 234 and 235). But the second member under 
this supposition, reduces to M, a finite number : hence, thd 
development cannot be made under that form. 

Again, assume, 

log y = My -\- iVy2 ^ p^s _|_ ^q^ 

U we make y = 0, we have 

log = that is, ± oo = 0, 

which is absurd, and therefore the development cannot be made 
under the last form. Hence, we conclude that. 

The logarithm of a number cannot he developed according to 
the ascending powers of that number. 

Let us write (1 -f- y), for y in the first member of the 
assumed development; we shall have, 

log (1 + y) =: My + Ay + Fy^ + Qy' + &c. - - (1), 

making y = 0, the equation is reduced to log 1=0, which does 
tiot present any absurdity. 

Since equation (1) is true for any value of y, we. may writa 
t for y; whence, 

log (l-{- z) = Mz + Nz^ + P^^ + Q^ -(- &c. - - . (2). 

Subtracting equation (2) from equation (1), member from mena- 
Der, we obtain, 

iog (t + y) - log (1 + ^) = M{ij -z)-\- A^(y2 - ^2) .f P(y3 _ ^) 

+ (2(y* - ^0 - - - (3). 



296 



ELEMENTS OF A.'.GEBRA. 



ICHAP. DL 



The second member of this equation is divisible by {y — z)^ 
let us endeavor to place the first member under such a form^ 
that it shall also be divisible by {y — z). We have, 

log (1 + y) - log (1 + z) = log ({-±^) = log (l + |-^). 

y — z 
But since can be regarded as a single quantity, we may 

X -f- z 

substitute it for y in equation (1), which gives, 

.- (. +HH)= -(m) +-eri-;F + -fr--;)"+- 

Substituting this development for its equal, in the first member 
of equation (3), and dividing both members of the resulting 
equation by (j/ — z), and we have, 

+ F{f + 2/2 + z^) + &c. 

Since this equation is true for all values of y and z, mAko- 
g =y, and there will result 
M 



1 + y 



= M+ 2Ky 4- 3Py2 j^ 4^Qy2 ^ ^j^^a _|_ ^q^ 



Clearing of fractions, and transposing, we obtain. 






y+SP 


3/2 + 4^ 


y' + ^B 


+ 2Ii 


4-3P 


-h4:Q 



y*+ &c. = 0, 



and since this equation is identical, we have, 
M— Mz=zO: whence, M=M: 



2N-\- M=0', whence, JST 



M 
2 ' 



3P+2iV^=0; whence, P=~ill=^; 

o 3 



4$ 4- SP = ; whence, ^ = — 



3P 
4 



4' 



.c<^ i-yra y^-\y\.'^\ 



CHAP. IX.j LOGARITHMIC SERIES. 297 

The law of the co-efficients in the development is evident; 

M 

the co-efficient of y" is + — , according as n is even or odd. 

Substituting these values for iV^, P, Q, &c., ii. equation (1), 
we find for the development of log (1 -\- y) ', 

M M M 

, log (1 -f 2/) = ^y - - y2 -f _y3 _ _ 2^4 _ &e. 

= 4_§H.{_5+lJ..,.e.)..(4). 

which is called the logarithmic series. 

Hence, we see that the logarithm of a number may be 
developed into a series, according to the ascending powers of 
a, number less than it by 1. 

In the above development, the co-efficients have all been de- 
termined in terms of M. This should be so, since M depends 
upon the base of the system, and to the base any value may be 
assigned. By examining equation (4), we see that, 

The expression for the logarithm of any number is composed of 
two factors, one dependent on the number, and the other on ths 
base of the system in which the logarithm is taken. 

The factor which depends on the base, is called the modulus 
of the system of logarithms. s^ • 

237* If ^^'e take the logarithm of 1 + y in a new system^ 
and denote it by / (1 + y), we shall have, 

l(l+y)^M'{y-f+^-^ + t-&o^. . (5), 

in which M' is the modulus of the new system. 

If we suppose y to have the same value in equations (4) and (5), 
and divide the former by the latter, member by member, we have 

1^-X) = J,; .W,(A,.WS3,) 

Z (1 + y) : log (I + y) : : J/' ; if ; hence, 
The logarithms of the same number, taken in two different systemti 
are to each other as the moduli of those systems. 

H 
■J 



298 ELEMENTS OF ALGEBRA. [CHAP. IX. 

238 1 Having shown that the modulus and base of a system 
of logarithms are mutuall} dependent on each other, it follows, 
that if a value be assigned to one of them, the corresponding 
Taluc of the other must be determined from it. 

If then, we make the modulus 

M' = 1, 
the base of the system will assume a fixed value. The system 
of logarithms resulting from such a modulus, and such a base, is 
CiiUed the Najjerian System. This was the first system known, 
and was invented by Baron Napier, a Scotch mathematician. 

If we designate the Naperian logarithm by /, and the loga- 
rithm in any other system by log, the above proportion becomes, 

Z(l+y) : log(l+y) : : 1 : J/; 
whence, M xl{l + y) = log (1 + y). 

Hence, we see that, 

The Naperian logarithm of any number, multiplied hy the modu- 
lus of any other system^ will give the logarithm of the same number 
in that system. 

The modulus of the Naperian System being 1, it is found most 
convenient to compare all other systems with the Naperian ; and 
hence, the modulus of any system of logarithms, is 

The number by which if the Naperian logarithm of ayiy 
number be multipjlied^ the product will be the logarithm of tlis 
same number in that system. 

239. Again, if x / (1 -f y) = log (1 -f y), gives 

Hl + y) = t3(l+l)., that, is, 

The logarithm of any number divided by the modulus of its 
ty stein, is equal to the Naperian logarithm of the same number. 

240. If we take the Naperian logarithm and make y = 1 
equation (5) becomes. 



CHAP. IX.] LOGAKITHMIC SERIES. 299 

a series tv'hich does not converge rapidly, and in which it would 
be necessary to take a great number of terms to obtain a near 
approximation. In general, this series will not serve for deter- 
mining the logarithms of entire numbers, since for every number 
greater than 2 we should obtain a series in which the terms 
would go on increasing continually. 

241. In order to deduce a logarithmic series sufficiently con 
verging to be of use in computing the Naperian logarithms 
of numbers, let us take the logarithmic series and make 
M'=^ 1. Designating, as before, the Naperian logarithm by l^ we 
shall have, 

;(l+y) = y-| + |-|^+|-&o. .-. (1). 
If now, we write in equation (1), — y for y, it becomes, 

Subtracting equation (2) from (1), member from member, 
we have, 

f(I+2/)-/(l-y) = 2(2/ + -f +^-' y +y + &c.)-. (3). 

But, 
/{I + y) - l{l -y) = I (j^) ; whence, 

If now we make = , we shall have, 

1 — y z ' ' 

(1 H- i/)z = (1 — y) (2 + 1), whence. 



22 -h 1 

Substituting these valurs in equatior (4), and observing that 
i(^-^) =l{2-\-l)-l3 we find, 



300 ELEMEJS'TS OF ALGEBRA. [CHAP. IX. 

/(. + l)-^. = 2(^-^ + ^^3+ ^^,+ &c.)(5). 
or, by transposition, 

Z(. +1) = fe + 2(^ + ^^, +_^ + &e.) (6), 

Let us make use of formula (6) to explain the method of 
computing a table of Naperian logarithms. It may be remarked, 
that it is only necessary to compute from the formula the 
logarithms of prime numbers; tho»e of other numbers may be 
found by taking the sum of the logarithms of their factors. 

The logarithm of 1 is 0. If now we make 5^ = 1, we can 
find the logarithm of 2 ; and by means of this, if we make 
£ = 2, we can find the logarithm of 3, and so on, as exhibited 
below. 

/1=0.--. = 0.000000 

IS = 0.G93147 + 2(l + 3L_ + ^ + ^...)^ 1.098612 

/4 = 2xZ2 - =1.386294 

^5 = 1.386294 + 2(1 + ^+^ + ^...)= 1.609437 

/6 = ?2+;3 =1.791759 

n = 1.791759 + 2(i + 3-l3-3+^,+ ...) = 1.945910 

/8 = M-f-^2 =2.079441 

/9 = 2x/3 =2.197224 

nO^Z5+^2 =2.302585 

<^x. &c. 

In like manner, we may compute the Naperian logarithms 
of all numbers. Other formulas may be deduced, which ar« 



CHAr. IX.] LOGARITHMIC SERIES. 301 

more rapidly converging than the one above given, but this 
serves to show the facility with which logarithms may be com- 
pjted. 

241*. We have already observed, that the base of the common 
system of logarithms is 10. We will now find its modulus. 
We have, 

^(1 + y) : log {1 + y) : . l:M (Art. 238). 

If we make y = 9, we shall have, 

nO: log 10 : : 1 : M. 

But the no = 2.302585093, and log 10=1 (Art. 228); 

hence, M = ^ ..^^^^-^^^ = 0.434294482 = the modulus of the 
common system. 

If now, we multiply the Naperian logarithms before found, by 
this modulus, we shall obtain a table of common logarithms 
(Art. 238). 

All that now remains to be done, is to find the base of the 
Naperian system. If we designate that base by e, we shall have 
(Art. 237), 

le : loge : : 1 : 0.434294482. 

But le = l (Art. 235): hence, 

1 : loge : : 1 : 0.434294482; 

hence, log e = 0.434294482. 

Bat as we have already explained the method of calculating 
the common tables, we may use them to find the number whose 
logarithm is 0.431294482, which we shall find to be 2.718281828 ; 
hence, 

e = 2.718281828 

We see frcm the last equation but one, that 

The modulus of the common system is equal to the common hga 
vithm of the Naperian base. 



802 ELEMENTS OF ALGEBRA. [CHAP. IX, 

Of Interpolation, 

242. When the law of a series is given, and several terms 
taken at equal distances are known, we may, by means of 
the formula, 

^ , n(n — \),, n(n — 1) (/J — 2) , . , 

T=a-\-nd,-ir \~-^d^ H' 1.2. 3^ +^^' ' * " (^)» 

already deduced, (Art. 209), introduce other terms between 
them, which terms shall conform to the law of the series. 
This operation is called interpolation. 

In most cases, the law of the series is not given, but only 
numerical values of certain terms of the series, taken at fixed 
intervals ; in this case we can only approximate to the law 
of the series, or to the value of any intermediate term, by 
the aid of formula (1). 

To illustrate the use of formula (1) in interpolating a term 
in a tabulated series of numbers, let us suppose that we have 
the logarithms of 12, 13, 14, 15, and that it is required to find 
the logarithm of 12^. Forming the orders of differences from 
the logarithms of 12, 13, 14 and 15 respectively, and taking 
the first terms of each, 

12 13 14 15 

1.079181, 1.113943, 1.146128, 1.176091, 

0.034762, 0.032185, 0.029963, 

- 0.002577, - 0.002222, 

+ 0.000355, 
we find d, = 0.034762, d,= - 0.002577, d^ = 0.000355. 

If we consider log 12 as the first term, we have also 

a = 1.079181 and n = —. 

lit 

Making these several substitutions in the formula, and ne- 
glecting the terms after the fourth, since they are hiappieciable. 
we find, 

T = a + —d,-- -d,^-—d,=z\og\^', 



CHAP. IX. J FORMULAS FOR INTEREST. 303 

or, by substituting for c?i, d^^ &c., their values, and for a its 
value, 

<». 1.079181 

Jrf, 0.017381 

irfj - - .... 0.000322 

^d^ • 0.000022 

Log 12^ .... 1.0969Q6 

Had it been required to find the logarithm of 12.39, we 
should have made 7i = .39, and the process would have been 
the same as above. In like manner we may interpolate terms 
between the tabulated terms of any mathematical table. 



INTEREST. 

243. The solution of all problems relating to interest, may 
b« greatly simplified by employing algebraic formulas. ' - 

In treating of this subject, we shall employ the following 
notation : 

Let p denote the amount bearing interest, called i\iQ principal ; 
r " the part of $1, which expresses its interest for 

one year, called the rate per cent.; 
t " the time, in years, that p draws interest ; 
i " the interest of p dollars for t years ; 
S " ^ 4- the interest which accrues in the time /. 

This sum is called the amount. 



Simple Interest. 

To find the interest of a sum p for t years^ at the rate r, and 
the amount then due. 



Since r denotes the part of a dollar which expresses its 
terest for a single year, the intprest of ^ dollars f(iF*-the sj 



m- 
ame 



804 ELEMENTS OF ALGEBRA. [CHAP. IX. 

iime will be expressed "by pr ; and for t years it will be t times 
as much : hence, 



i —ptr - - - . - . 


- (1); 


and for the amount due, 




S = p-^ptr=zp{l-{.tr) . 


■ (2). 


EXAMPLES. 





1. What is the interest, and what the amount of $365 for three 
years and a half, at the rate of 4 per cent, per annum. Here, 

p — $365 ; 

i = 3.5 ; 

i = ptr — 365 X 3.5 X 0.04 = $51,10 : 
hence, /S' = 365 + 51,10 = $416,10. 



^ 



Present Value and Discount at Simple Interest 



The present value of any sum 5, due t years hence, is the prin- 
cipal j9, which put at interest for the time ^, will produce the 
amount >S'. 

The discount on any sum due t years hence, is the difference 
between that sum and the present value. 

To find the present value of a sum of dollars denoted hy S, di^ 
i years hence, at simple interest, at ike rate r; also, the discount. 

We have, from formula (2), 

S =p -\- ptr ; 

and since p is the principal which in t years will produce the 
Bum S, we have^ 



V 



.oli 



f" - ^-rh ^'y- 



CHAl'. IX.1 FORMULAS FOR INTEREST. 305 

and for the discount, which we will denote by i>, we have 
_ ^ S Sir 

1. Required the discount on $100, due 3 months hence, at the 
rate of 5J per cent, per annum. 

^=$100 =$100, 

t = 3 months = 0.25. 



Hence, the present value p is 
S 100 



$98,643 



^ l + tr 1 + .01375 
hence, J) =:S-p =100- 98,64S = $1,357. 

Compound Interest, 

Compound interest is when the interest on a sum of money- 
becoming due, and not paid, is added to the principal, and": 
the interest then calculated on this amount as on a new 
principalc 

To fiyid the amount of a sum p placed at interest for t yearSy. 
compound interest being allowed annually at the rate r. 

At the end of one year the amount will be, 

S=p-\-pr = 2o{l-\-r). 

Since compound interest is allowed, this sum now becC^riea 
the principal, and hence, at the end of the second year, the 
amount will be, 

S^ =j^(l -f r) +^r(l + r) = p(\ -f r)\ 

Regard p{\ + r)"^ as a new principal ; we have, at the end 
of the third year, 

S''.= p{\+ry+pr{l-[-ry=p{\ + rY', 
20 



306 ELEMENTS OF ALGEBRA. LCHAP. IX, 

asd at the end of t years, 

s=:p{y^-TY .... (5). 

And from Articles 230 and 232, we have, 

log S = logp + t log (1 -^ r) ; 

and if any three of the four quantities aS, p, /, and r, are givei\ 
the remaining one can be determined. 

Let it be required to find the time in which a sum p will 
double itself at compound interest, the rate being 4 per cent, 
per annum. 

We have, from equation (5), 

S =p{l + ry. 

But by the conditions of the question, 
S=2p=p{l-{-ry: 
hence, 2 = (l+r)». 

and - ^Qg^ _ 0-301030 

*^ ~ log (l+r) ~ 0.017033* 

= 17.673 years, 

= 17 years, 8 months, 2 days. 

To find the Discount 

The discount being the difference between the «um S and p^ 
we have, 



CHAPTER X. 



GENERAL THEORY OF EQUATIONS. 



244. Every equation containing but one unknown quantity 
which is of the m** degree, m being any positive whole number, 
may, by transposing all its terms to the first member and divid- 
ing by the co-efficient of a:*", be reduced to the form 

In this equation P, Q^ , . , . T^ U, are co-efficients in the 
most general sense of the term ; that is, they may be positive 
or negative, entire or fractional, real or imaginary. 

The last term JJ is the co-efficient of a;^, and is called the 
absolute term. 

If none of these co-efficients are 0, the equation is aiid to be 
complete; if any of them are 0, the equation is said to be 
incomplete. 

In discussing the properties of equations of the m*^ degree, 
mvolving but one unknown quantity, we shall hereafter suppose 
them to have been reduced to the form just given. 

245. We have already defined the root of an equation (Art. 77) 
to be any expression^ which, when substituted for the unknown 
quantity in the equation, will satisfy it 

We have shown that every equation of the first degree has 
one root, that every equation of the second degree has two 
roots; and in general, if the two members of an equation are 
equal, they must be so for at least some one value of the 



308 ELEMENTS OF ALGEBRA. [CHAP. X. 

unknown quantity, either real or imaginary. Such value of the 
unknown quantity is a root of the equation : hence, we infer, tbit 
tvery equation^ of whatever degree^ has at least one root. 

We shall now demonstrate some of the principal properties 
of equations of any degree whatever. 

First Property. 
246. In every equation of the form 

^m ^ p^m-\ _^ Qxm-2 ^ _ _ + 2^2; + U= 0, 

if & is a root, the first member is divisible by x — a ; and con 
versely, if the first member is divisible by x — a, a is a root of 
the equation. 

Let us apply the rule for the division of the first member 
by X — a, and continue the operation till a remainder is found 
which is independent of x ; that is, which does not contain x. 

Denote this remainder by B and represent the quotient found 
by Q', and we shall have, 

^m _|_ p^m-i ^ ^ ^ ^ ^ Tx+ U= Q\x — a)+R. 

Now, since by hypothesis, a is a root of the equation, if we 
substitute a for x, the first member of the equation will reduce to 
zero ; the term Q'{x — a) will also reduce to 0, and consequently, 
we shall have 

R = 0. 

But since B does not contain x, its value will not be afieeted 
by attributing to x the particular value a : hence, the remainder 
R is equal to 0, whatever may be the value of ar, and conse- 
quently, the first member of the equation 

^m _|_ p^m-\ ^ Q^m-2 , , , , ^ Tx -{- U = 0, 

Is exactly divisible by x — a. 

Conversely, if a; —- a is an exact divisor of the first member 
of the equation, the quotient Q' will be exact, and we shall have 
R = 0: hence, 

^m ^ pa-m-l . . _ 4. Tip ^. I/- ^'(^ _ a). 



CHAP. X.] THEORY OF EQUATIONS. 309 

If now, we suppose x = a, the second member will reduce to 
zero, consequently, the first will reduce to zero, and hence a will 
be a root of the equation (Art. 245). It is evident, from the 
nature of division, that the quotient Q' will be of the form 

^m-l _|. p'^m-2 + B'X+ U' = 0. 

247. It follows from what has preceded, that in order to dis- 
cover whether any polynomial is exactly divisible by the bino- 
mial X — a, it is sufficient to see if the substitution of a for J 
will reduce the polynomial to zero. 

Conversely, if any polynomial is exactly divisible by x — a, 
then we know, that if the polynomial be placed equal to zero, 
a will be a root of the resulting equation. 

The property which we have demonstrated above, enable» us 
to diminish the degree of an equation by 1 w^hen we kLow 
one of its roots, by a simple division ; and if two or m7.»re 
roots are known, the degree of the equation may be still fur'^her 
diminished by successive divisions. 

EXAMPLES. 

1. A root of the equation, 

x^ - 25a;2 4- QOx _ 36 = 0, 
b 3 : what does the equation become when freed of this cot ? 
a;* - 252:2 + 60a; - 36 lb _3 



X* 


— 


Sx^ 


^3 


+ 3a 




+ 


3^3- 
3^3 


-25a;2 
- 9a;2 








- 16a;2 + QOx 

— 16a;2 + 48a; 





12a; - 36 
12a; - 36 
Ans. x^ + 3a:2 _ iq-^ .^ 12 - 0. 
2. Two roots of the equation, 

a;* - 12a;3 4. 43^2 _ gg^ + 15 = 0, 
are 3 and 5 : what does the equation become when freed ^*f 
them ] Ans. a;2 — 4a; + 1 = C 



810 ELEMENTS OF ALGEBRA. [CHAP. X. 

3. A root of the equation, 

is 1 : what is the reduced equation 1 

Ans. a;2 — 5a; + 6 = 0. 

4. Two roots of the equation, 

4:x^ — 14a-3 — 5x^ + 31a; + 6 = 0, 
are 2 and 3 : find the reduced equation. 

Ans. 4a;2 -f 6a; + 1 = 0. 

Second Property. 

248 • Every equation involving hut one unknown quantity, has 
zs many roots as there are units in the exponent which denotes 
its degree, and no more. 

Let the proposed equation be 

Since every equation is known to have at least one root 
(Art. 245), if we denote that root by a, the first member will 
be divisible by x — a, and we shall have the equation, 

j.m 4. p^m-\ + . _ _ (^ _ a) (^»n-l _j_ p/-^;m-2 +...).... (1). 

But if we place, 

Xm-\ 4- p/a;'"-2 + . . . :rz 0, 

we obtain a new equation, which has at least one root. 

Denote this root by 5, and we have (Art. 246), 

a;m-i _|_ p^x'^-2 4. . _ = (a; — 6) (a;'"-2 + P^'x"^ +...)• 

Substituting the second member, for its value, in equation 
(1), we have, 
af -f Pa;*"-! ^ , , , = {x — a) {x ~h) (a;'»-2 + P'' x^"^ + ..-)• (2). 

Reasoning upon the polynomial, 

Xmr~2 4_ p//^»n-3 4. . . ., 

fts upon the preceding polynomial, we have 

gff^2 4. p//a;m-3 4. . _ _ (3; _ f) [x^"^ + p'^^x''^ +...)» 

and by substitution, 

««-j-Pa;'^i.f . . . — {x ~ a){x --h){x — c) {x-^'^-\-P'"x'^ - - • (3). 



CHAP. X.I THEORY OF EQUATIONS. 311 

By continuing this operation, we see that for each bii»omial 
fector of the first degree with reference to a:, that we separate, 
the degree of the polynomial factor is reduced by 1 ; therefore, 
after m — 2 binomial factors have been separated, the polynomial 
factor will become of the second degree with reference to x^ 
which can be decomposed into two factors of the first degree 
(Art. 115), of the form x — k, x — /. 

Now, supposing the m — 2 factors of the first degree to have 
already been indicated, we shall have the identical equation, 

x"^ -1- Px"^'- f . . ={x — a) {x — h){x — c).,{x — lc){x — l) = (^'^ 

from which we see, that the first member of the proposed equation 
may be decomposed into m binomial factors of the first degree. 

As there is a root corresponding to each binomial factor of 
the first degree (Art. 246), it follows that the m binomial factors 

of the first degree, x — a^ x — b, x — c , give the m roots, 

a, 6, c . . ., of the proposed equation. 

But the equation can have no other roots than a, 5, c . . . yt, /. 
for, if it had a root a^, different from a, 6, c ..../, it would 
have a divisor x — a^, different from x — a, x — 6, x — c...x — /, 
which is impossible ; therefore, 

Every equation of the m** degree has m roots, and can haw 
no more. 

249. In equations which arise from the multiplication of equal 
factors, such as 

{x - ay {x - by {x - cy {x-^d) = o, 

the number of roots is apparently less than the number of units 
in the exponent which denotes the degree of the equation. But 
this is. not really so; for the above equation actually has ten 
roots, four of which are equal to a, three to 6, two to c, and 
one to d. ' 

It is evident that no quantity a% different from a, &, c, rf, 
can verify the equation ; for, if it had a root a^, the first mem- 
ber would be divisible hjx — a\ which is impossible. 



312 ELEMENTS OF ALGEBRA. FCHAP. X. 

* Consequence of the Second Property. 

250. It has been shown that the first member of every equa- 
tion of the m^^ degree, has m binomial divisors of the first 
degree, of the form 

ic — a, X — b, X — Cy , , . X — k, x — I. 

If we multiply these divisors together, two and two, three and 
three, &c., we shall obtain as many divisors of the second, 
third, &;c. degree, with reference to x, as we can form different 
combinations of m quantities, taken two and two, three and three, 
<kic. Now, the number of these combinations is expressed by 

m — \ m — 1 m — 2 ,. 

m.— -^— , m.— ^— .— ^— . . . (Art. 132); 

hence, the proposed equation has 

m — \ 

divisors of the second degree ; 

m — \ m — 2 



2 3 

divisors of the third degree ; 

m — \ m — 2 



m . 



2 3 4 

divisors of the fourth degree ; and so on. 

Composition of Equations. 

251. If we resume the identical equation of Art. 248, 
««+PiC'"-i + Qx"^-"^ . . . + U = {x—a) {x —b)(x — c) . . . (a?— t) . . . 
and suppose the multiplications indicated in the second member 
to be performed, we shall have, from the law demonstrated in 
article 135, the following relations : 

P =z — a-h —c — ...—k—l, or — P = a+6 + c+ ..-^-k+l, 
Q =z ah -{- ac -\- be •{•.... ak -\- hi, 
E = -^ abc — abd —bed ... — iki, or — P z=.ahc -j- o-bd + . . .+ ikl^ 

U= ±: abed .... iklf or zt U = abc . . . ikl. 



CHAP. X.] COMPOSITION OF EQUATIONS. .S13 

The double sign has been placed before the product of a, 5, c, &c. 
in the last equation, since the product — a X — ^X — c . . X — l, 
will be plus when the degree of the equation is even, and minus 
when it is odd. 

By considering these relations, w^e derive the following conclu- 
sions with reference to the values of the co-efficients: 

1st. The co-efficient of the second term, with its sign changed, is 
equal to the algebraic sum of the roots of the equation. 

2d. The co-efficient of the third term is equal to the sum of the 
diffiereni products of the roots, taken two in a set. 

3d. 21ie co-efficient of the fourth term, with its sign changed, is 
equal to the sum of the different products of the roots, taken three 
in a set, and so on. 

4th. The absolute term, with its sign changed when the equation 
ts of an, odd degree, is equal to the continued product of all the 
roots of the equation. 

Consequences. 

1. If one of the roots of an equation is 0, there will be 
no absolute term; and conversely, if there is no absolute term, 
?ne of the roots must be 0. 

2. If the co-efficient of the second term is 0, the numerical 
sum of the positive roots is equal to that of the negative roots. 

8. Every root will exactly divide the absolute term. 

It will be observed that the properties of equations of the 
second degree, already demonstrated, conform in all respects to 
the princ pies demonstrated in this article. 

EXAMPLES OF THE COMPOSITION OF EQUATIONS. 

1. Find the equation whose roots are 2, 3, 5, and — 6. 
We have, from the principles already established, the equation 
(x - 2) (a; - 3) (a; - 5) (a; -f G) = ; 
whence, by the application of the preceding principles, we obtain 
the equation, 

X* - 4x-3 - 29a;2 4- 156a; - ISO = 0. 



S14 ELEMENTS OF ALGEBRA. LCHAP X 

2. What is the equation whose roots are 1, 2, and — 3 1 

Ans. x^ -7x + Q = 0. 

5J. What is the equation whose roots are 3, — 4, 2-4- ^/3, 
nd 2 -^3"? Ans. x* — Sx^ — I5x^ + 49a; ~ 12 = 0. 

4. What is the equation whose roots are 3+,V5, 3 — y^, 
and — 6? Ans. x^ — 32a; + 24 = 0. 

5. What is the equation whose roots are 1, — 2, 3, — 4, 5, 
and — 6 ? 

Ans. x^ + Sx^ — 4:lx^ — 87a;3 + 400a;2 + 444a; — 720 = 0. 



6. What is the equation whose roots are .... 2 + -/ — 1, 
2-y^^^, and -31 Ans. z^ —x^ — 7x + 15 = (^ 

Greatest Common Divisor. 

252. The principle of the greatest common divisor is of fre- 
quent application in discussing the nature and properties of 
equations, and before proceeding further, it is necessary to inves- 
tigate a rule for determining the greatest common divisor of two 
or more polynomials. 

The greatest common divisor of two or more polynomials is 
the greatest algebraic expression, with respect both to co-efficienta 
and exponents, that will exactly divide them. 

A polynomial is prime, when no other expression except I 
will exactly divide it. 

Two polynomials are prime with respect to each other, when 
they have no common factor except 1. 

253. Let A and B designate any two polynomials arranged 
with reference to the same leading letter, and suppose the 
polynomial A to contain the highest exponent of the leading 
letter. Denote the greatest common divisor of A and B by Z>, 
und let the quotients found by dividing each polynomial by B 



CHAP. X.] GREATEST COMMON" DIVISOK. 315 

be represented by A^ and B' respectively. We shall then have 
the equations, 

1 = ^'. and 1 = ^; 

whence, A — A y. D and B — B' y, D, 

Now, D contains all the factors common to A and B. For, 
if it does not, let us suppose that A and B have a common 
factor d which does not enter i>, and let us designate the quo- 
tients of A' and B\ by this factor, by A' and B'\ We shall 
tJien have, 

A=^A'\d.D and B = B'\d.D', 

or, by division, 

^ =A'' and -^ = B'\ 



d.D d.D 

Since A^' and j5" are entire, .both A and -S are divisible by 
d . D, which must be greater than i>, either with respect to its 
co-efficients or its exponents ; but this is absurd, since, by 
hypothesis, D is the greatest common divisor of A and B, 
Therefore, D contains all the factors common to A and B. 

Nor can D contain any factor which is not common to A 
and B. For, suppose D to have a factor d^ which is not con 
tained in A and B, and designate the other factor of D by Z>' ; 
we shall have the equations, 

A = A\d\2)' and B = B\ d' . IT ; 

or, dividing both members of these equations by d\ 

— = A\D' and ?- = B\D\ 
a d' 

Now, the second members of these two equations being en- 
tire, the first members must also be entire ; that is, both A 
and B are divisible by d' and therefore the supposition that 
d' is not a common factor A A and B is absurd. Hence, 

1st. The greatest common divisor of two polynomials contains 
all the factors common to the polynoiiials, and does not cx)ntain 
any other factors. 



816 ELEMENTS OF ALGEBRA. LCHAP. X. 

254. If, now, we apply the rule for dividing A by B, and 
continue the process till the greatest exponent of the leading 
letter in the remainder is at least one less than it is in the 
polynomial B, and if we designate the remainder by i2, and 
the quotient found, by Q, we shall have, 

A = By Q-hB - - - - (1). 

If, as before, we designate the greatest common divisor of 
A and B by i>, and divide both members of the last equation 
by it, we shall have, 

A B ^.R ■ 

Now, the first member of this equation is an entire quantity, 

and so is the first term of the second member ; hence — 

must be entire; which proves that the greatest common divisor 
of A and B also divides B. 

If we designate the greatest common divisor of B and R by 
ly^ and divide both members of equation (1) by it, we shall have, 

Now, since by hypothesis i/ is a common divisor of B and 
R^ both terms of the second member of this equation are 
entire ; hence, the first member must be entire ; which proves 
that the greatest common divisor of B and i2, also divides A, 

We see that B\ the greatest common divisor of B and R 
cannot be less than i), since D divides both B and R\ nor can 
i>, the greatest common divisor of A and B^ be less than D\ 
because D' divides both A and B ; and since neither can be less 
than the other, they must be equal ; that is, D = D\ Hence, 

2d. The greatest coramon divisor of two polynomials^ is the same 
as that betiveen the second polynomial and their remainder after 
division. 

From the principle demonstrated in Art. 253, we see that wo 
may multiply or divide one polynomial by any factor that ia 



CHAP. X.1 GREATEST COMMON DIVISOR. 317 

not contained in the other, without affecting their greatest com- 
mon divisor. 

255, From the principles of the two preceding articles, we 
deduce, for finding the greatest common divisor of two poly- 
nomials, the following 

RULE. 

J. Suppress the monomial factors common to all the terms of the 
first polynomial ; do the same with the second polynomial ; and if 
the factors so suppressed have a common divisor^ set it aside, as 
forming a factor of the common divisor sought, 

II. Prepare the first polynomial in such a manner that its first 
term shall be divisible by the first term of the second polynomial, 
both being arranged with reference to the same letter : Apply the 
rule for division, and continue the process till the greatest exponent 
of the leading letter in tj^e remainder is at least one less than it is 
in the second polynomial. Suppress, in this remainder, all the 
factors that are common to the co-efficients of the different powers 
of the leading letter ; then take this result as a divisor and the 
second polynomial as a dividend, and proceed as before. 

III. Continue the operation until a remainder is obtained which 
will exactly divide the preceding divisor ; this last remainder, mul- 
tiplied by the factor set aside^ will be the greatest common divisor 
sought ; if no remainder is found which will exactly divide the 
preceding divisor, then the factor set aside is the greatest common 
divisor sought. 

EXAMPLES. 

1. Find the greatest common divisor of the polynomials 
a3 _ a^b + 3a62 _ 3^3, and a^ — 5ab + Ab^. 
First Operation. Second Operation. 



a3_a26-f3a62_363 



4a26 — ab^ — Sb^ 



|a2 6ab + 462 



a 4-46 



1st rem. 19a62 — 1963 
or, 1962 (a -6). 

Hence, c — 6 is the greatest common divisor. 



a2 — 6ab + 462 



_ 4ab -f 46' 



Ab 



0. 



318 ELEMENTS OF ALGEBRA. [CHAP. X 

We begin by dividing the polynomial of the highest degree 
by that of the lowest ; the quotient is, as we see in the above 
table, a + 46, and the remainder lOafi^ — 19^3^ 

But, 19a62 _ 1963 ^ 1952 (^ _ b). 

Now, the factor 196^, will divide this remainder without dividing 

a2 _ 5^5 ^ 452 . 

hence, th3 factor must be suppressed, and the question is reduced 
to finding the greatest common divisor between 
a? — bah + 46^ and a — b. 
Dividing the first of these two polynomials by the second, there 
is an exact quotient, a — 4b; hence, a — b is the greatest com- 
mon divisor of the two given polynomials. To verify this, lei 
each be divided by a — b. 

2. Find the greatest common divisor of the polynomials, 

3a5 - 5a362 4. 2a6* and 2a' — ZaW + 6*. 
We first suppress a, which is a factor of each term of the 
first polynomial : we then have, 

3a* - baW + 26* II 2^* - 3a262 + 6*. 
We now find that the first term of the dividend will not con- 
tain the first term of the divisor. We therefore multiply the 
dividend by 2, which merely introduces into the dividend a 
factor not common to the divisor, and hence does not affect 
the common divisor sought. We then have, 

6a* — 10a262 + 46* i|2a* — 3a262 -f 6* 



(6a^— 9a262 + 364 



— a262+ 6* 

_ 62 («2 _ J2)^ 

We find afler division, the remainder — a252 ^ J* which we 
put under the form — 6^ [a? — b^). We then suppress — b\ 
and divide. 



2a* - 3a262 -f- 6* 


1 a2 - 62 


2a* - 2a262 


2a2 - 62 



— a262 -I- 6* 

— a262 + 6*. 

Hence, a^ — 6^ is the greatest ccmmon divisor. 



CHAP. X.J GREATEST COMMON DIVISOR. 819 

3. Let it be required to find the greatest common divisor 
between the two polynomials, 

— 3Z>3 -f 3a62 _ a?b + a^, and 46^ — bah + a\ 
First Operation, 



— \2U^ + I2ab^ - Aa% + 4a3 



!jst rem. - - - Zab"^ — a^ + 4a3 
— 12a62-4a26 + IQa^ 



Ab^ — 5a6 4- a2 



3^>, - 3a 



2d rem. • - — 19a^ -i- 19a^ 

or, 19a2(-6 + a). 

Second Operation. 



462 _ 5a5 _|_ ^2 



ab 



- 6 + a 



46 + a 



0. 
Hence, — 6 + a, or a — b, is the greatest common divisor 
In the first operation we meet with a difficultj in dividing the 
two polynomials, because the first term of the dividend is not 
exactly divisible by the first term of the divisor. But if w© 
observe that the co-efficient 4, is not a factor of all the terms 
of the polynomial 

and therefore, by the first principle, that 4 cannot form a part 
of the greatest common divisor, we can, without affecting this 
common divisor, introduce this factor into the dividend. This 

gives, 

- 1263 + 12a62 _ 4a26 + 4a3, 

and then the division of the terms is possible. 

Effecting this division, the quotient is — 36, and the re 

mainder is, 

— 3a62 _ a^ + 4a^, 

As the exponent of 6 in this remainder is still equal to 

that of 6 in the divisor, the division may be continued, b^ 

multiplying this remainder by 4, in order to render the division 

of the first term possible. This done, the remainder becomes 

- 12a62 - 4a26 -{- 16a3 ; 



820 ELEMENTS OF ALGEBRA. [CHAP. X. 

which, divided by 4&2 _ ^ah -f a^, gives the quotient — 3a, 
which should be separated from the first by a comma, having 
no connexion with it. The remainder after this division, is 

— Ida^ + 19a3. 

Placing this last remainder under the form 19a^ (— & + a), 
and suppressing the factor 19a?, as forming no part of the com- 
mon divisor, the question is reduced to finding the greatest 
common divisor between 

4Z»2 — 5ab + a^ and — b -\- a. 

Dividing the first of these polynomials by the second, we 
obtain an exact quotient, — 46 + a : hence, — b -{- a, or a — 6, 
is the greatest common divisor sought. 

256. In the above example, as in all those in which the 
exponent of the leading letter is greater by 1 in the dividend 
than in the divisor, we can abridge the operation by first mul- 
tiplying every term of the dividend by the square of the co- 
efficient of the first term of the divisor. We can easily see 
that by this means, the first term of the quotient obtained will 
contain the first power of this co-efficient. Multiplying the 
divisor by the quotient, and making the reductions with the 
dividend thus prepared, the result will still contain the co-efficient 
as a factor, and the division can be continued until a remainder 
is obtained of a lower degree than the divisor, with reference 
to the leading letter. 

Take the same example as before, viz. : 

— 3^3 4- 3a52 — a^ -f a^ and 462 _ 5^5 _j_ a\ 
and multiply the dividend by 4^ = 16 ; and we have 

First Operation. 

— 4863 + 48a62 _ 16a26 4- I6a3 ||462^-5(^M^ 
— 12a62 — 4a26 + 16a-^ - 126 - 3a 



1st remainder, — 19a26 -|- 19a3 

or, 19a2(-6 + a). 



CHAP. X.] GREATEST COMMON DIVISOR. 321 



Second Operation. 



462 _ 5oj + «2 



ab + a^ 



— b-{-a 



46 + a 



*2(3 remainder, — 0. 

When the exponent of the leading letter in the dividend 
exceeds that of the same letter in the divisor by two, three,. 
&c., multiply the dividend by the third, fourth, &c. povrer of 
the co-efficient of the first term of the divisor. It is easy to- 
see the reason of this. 

257. It may be asked if the Piippressiion of the factors, com 
mon to all the terms of one of the remainders, is absolutely 
necessary^ or whether the object is merely to render the opera- 
tions more simple. It will easily be perceived that the suppres- 
sion of these factors is necessary ; for, if the factor lOa^ was not 
suppressed in the preceding example, it would be necessary to 
multiply the whole dividend by this factor, in order to render 
its first term divisible by the first term of the divisor ; but, 
then, a factor would be introduced into the dividend which is 
also contained in the divisor ; and, consequently, the required 
greatest common divisor would contain the factor IQa^ which, 
should form no part of it. 

258. For another example, let it be required to find the 
greatest common divisor of the two polynomials, 

a* + Za% + 4a262 _ (Sa¥ + 26^ and 4.a% + 2a62 _ 2b\ 

ur simply of, 

a* + Za^b + 4a262 — Qab^ -f 26* and 2a2 -\- ab — 62, 

since the factor 26 can be suppressed, being a factor of the 
second polynomial and not of the first. 

First Operation. 
8a* -i- 2ia36 + 32^262 — 48a63 -f 166* 



2a2 + a6 - 62 
+ 20a36 -f 36a262 - 48a63 + 166* 4a2 + 10a6 -f 136^ 



4- 26^262 _38a63+ 166* 

1st remainder, — b\ab^ + 296* 

or, - 63(51a - 296), 



322 ELEMENTS OF ALGEBRA. ICHAP. X. 

Second Operation. 

Multiply by 2601, the square of 51. 

5202a2 + 2601a6 - 260152 | | 5ig - 29^ 

5202a2 - 295Sa6 j 102a -f 1095 

1st remainder, + 5559a6 — 26015^ 
5559a5- 316162 



2d remainder, + 560^2. 

The exponent of the letter a in the dividend, exceeding thai 
of the same letter in the divisor, by tivo^ the whole dividend 
is multiplied by 2- = 8. This done, we perform the division^ 
and obtain for the first remainder, 

- 51a63 + 295*. 

Suppressing — 5^, this remainder becomes 51a — 295 ; and 
the new dividend is 

2a2 ^ ah — 52. 

Multiplying the dividend by (51)2 == 2601, ^^-^ effecting the 
division, we obtain for the second remainder 4- 56052. Now, it 
results from the second principle (Art. 254), that the greatest 
common divisor naust be a factor of the remainder after each 
division; therefore it should divide the remainder 56052. g^,^ 
this remainder is independent of the leading letter a : hence, if 
the two polynomials have a common divisor, it miust be inde- 
pendent of a, and. will consequently be found as a factor in the 
co-efTicients of the different powers of this letter, in each of the 
proposed polynomials. But it is evident that the co-efRcients of 
these powers have not a common factor. Hence, the tivo given 
polynomials are prime with respect to each other. 

259. The rule for finding the greatest common divisor of two 
polynomials, may readily be extended to three or more poly 
nomials. For, having the polynomials A^ B, C, D, &:c., if we 
hnd the greatest common divisor of A and £, and then the 
greatest ' common divisor of this result and (7, the divisor so ob 



CHAP. X.] GREATEST COMMON DIVISOR. 323 

tained will evidently be the greatest common divisor of A, B^ 
and C\ and the same process may be applied to the remaining 
polynomials. 

260. It often happens, after suppressing the monomial factors 
common to all the terms of the given polynomials, and arranging 
the remaining polynomials with reference to a particular letter, 
that there are polynomial fictors common to the co-efficients of 
the different powers of the leading letter in one or both poly- 
nomials. In that case we suppress those factors in both, and if 
the suppressed factors have a common divisor, we set it aside, as 
forming a factor of the common divisor sought. 

EXAMPLE. 

Let it be required to find the greatest common divisor of the 
two polynomials 

a^d? — c?dP' — a^c^ + c\ and AaH — 2ac^ -f 2c^ — 4acd. 

The second contains a monomial factor 2. Suppressing it, 
and arranging the polynomials with reference to c?, we have 

(a2 — c2) c^2 _ ^2^2 ^ c\ and (2a2 _ 2ac) d ~ ac"^ + c\ 

By considering the co-efficients, a^ — c^ and — ah^ + c*, in the 
first polynomial, it will be seen that — a^c^ -j- c* can be put under 
the form — c'^(a'^ — c'^): hence, a^ — c^ is a common factor of the 
co-efficients in the first polynomial. In like manner, the co-effi- 
cients in the second, 2a2 _ 2ac and — ac^ -\- c^, can be reduced 
to 2a{a — c) and — c'^{a — c) ; therefore, a — c is a common 
factor of these co-efficients. 

Comparing the two factors a^ —c^ and a — c, we see that the 
last will divide the first ; hence, it follows that a — c is a com- 
mon factor of the proposed polynomials, and it is therefore a 
factor of the greatest common divisor. 

Suppressing a^ — c^ in the first polynomial, and a — c in the 
second we obtain the two polynomials, 

<j?2 _ c2 and 2ad — d^. 



824 ELEMENTS OF ALGEBRA, LCHuU», X, 

to wnich the ordinary process may be applied. 



rf2_c2 


2ad — c2 


4aV2 - 4a2c2 


2aJ 4- c2 


+ 2ac2cZ — 4a2c2 




- 4a2c2 + c*. 





After having multiplied the dividend by 4a 2, and performed 
the division, ve obtain a remainder — 4a2c2 -f c*, independent of 
tlie letter d : hence, the two polynomials, d^ — c^ and 2a(/ — c2, 
are prime -with respect to each other. Therefore, the greatest 
common divisor of the -proposed polynomials is a — c. 

261. It sometimes happens that one of the polynomials cou 
tains a letter which is not contained in the other. 

In this case, it is evident that the greatest common divisor is 
independent of this letter. Hence, by arranging the polynomial 
which contains it, with reference to this letter, the required com- 
mon divisor will be the same as that which exists hetvjeen the co- 
efficients of the different powers of the jnincipal letter and the 
second polynomial. 

By this method we are led, it is true, to determine the great 
est common divisor between three or more polynomials. But 
tliey will be more simple than the proposed polynomials. It 
often happens, that some of the co-efficients of the arranged 
polynomial are monomials, or, that we can discover by simple 
inspection that they are prime with respect to each other ; and, 
h: this case, we are certain that the proposed polynomials are 
prime with respect to each other. 

Thus, in the example of the last article, after having suppresseil 
the common factor a — c, which gives the results, 

^2 _ c2 and 2ad - c2, 

we know immediately that these two polynomials are prime with 
respect to each other ; for, since the letter a is contained in the 
second and not in the first, it follows from what has just been said, 
that the common divisor must be contained in the co-efficients 2'i 



CHAP. X.J GREATEST COMMON DIVISOR. 825 

and — c^ ; but these are prime with respect to each 3ther, and 
consequently, the expressions cp — c^ and 2ac/ — c^, are also prime 
with respect to each other. 

Let it be required to find the greatest common divisor of the 
tw^o polynomials, 

Zhcq + SO??ip +\Shc + bmpq, 
and, Aadq — 42^ + 2-iad — '^fgq. 

Now, the letter b is found in the first polynomial and not in 
the second. If then, w^e arrange the first with reference to i, 
we have, 

{Zcq -\-\Sc)h-\- ZOmp + bmpq, 

and the required greatest common divisor will be the same as 
that which exists between the second polynomial and the two 
co-efficients of 5, which are, 

ocq -[-18c and 30 mp -f- ^mpq. 

Now, the first of these co-efficients can be put under the form 
Zc[q -j- 6), and the other becomes bmp{q -{- 6) ; hence, g' -|- 6 is 
a common factor of these co-efficients. It will therefore be 
sufficient to ascertain whether $' + 6 is a factor of the second 
polynomial. 

Arranging this polynomial with reference to g, it becomes 
{4ad-'7fg)q-42fg-\-24ad', 

and as the second part, 24:ad — 42fg = 6{4.ad — 7f(/), it follows 
that this polynomial is divisible by g' -|- 6, and gives the quotient 
4ac? — 7fg. Therefore, <7 -f 6 is the greatest common divisor of 
the proposed polynomials^ 

EXAMPLES. 

1. Find the greatest common divisor of the two polynomial* 

6a;5 _ 4.f4 - Ux^-Sx^-Sx-l, 

and 4x* -h 2x^ — \Sx^ -{- Sx — 5. 

Ans 2x^ — 42'2 -J- ar — 1 



326 ELEMENTS OF ALGEBRA. I CHAP. X, 

2. Find the greatest common divisor of the polynomials 

20:^6 - 12a;^^ + 16^;* - l^x^ + Ux^ - 15a; + 4, ' 
and 15a;* - 92:3 _^ 47^2 _ 21a; + 28. 

Ans. bx"^ — 3a; -f 4. 

3. Find the greatest common divisor of the two polynomiaia 

5a*62 4. 2aW + ca? — Sa^^ -f bca, 
and a^ + ba^d — aW + ba^bd. 

Ans a'^ -{- ab. 

Transformation of Equations. 

262. The object of a transformation, is to change an equation 
from a given form to another, from which we can more readily 
determine the value of the unknown quantity. 

First. 

To change a given equation involving fractional co-efficien ts to another 
of the same general form ^ hut having the co-efficients of all its terms entire 
If we have an equation of the form 

y 

and make a; = — ; 

in which y is a new unknown quantity, and Jc entirely arbitrary ; 
we shall have, after substituting this value for a;, and multiplying 
every teim by ^'"j 

ym 4. p]cy^-\ 4-. QBym-2 -|_ RJ^^ym-2 + . . . + Tk'^-'^y + fTX''" =. 0, 

an equation in which the co-efficients of the different powerc of 
y are equal to those of the same powers of x in the given equa- 
liv)n, multiplied respectively by ^o, A-i, k"^, k^, ^*, &:c. 

It is now required to assign such a value to Jc as will make 
the CO efficients of the different powers of y entire. 

To irxititrate, let us take, as a general example, the equation 



CHAF X.] TKANSFOKMATION OF EQUATIONS. 327 

which becomes, after substituting -j- for ar, and multiplying by k\ 
ah , ck"^ „ eh^ . qJc* 

Now, there may be two cases — 

1st. Where the denominators 6, c?, /, A, are prime with respect 
to each other. In this case, as k is altogether arbitrary, take 
k =: bdfh, the product of the denominators^ the equation will then 
become, 

y^ ._f_ adfh . 2/3 + c6V/2/i2 . 2/2 _|. ebWph^ . ?/ + gbH'^fh^ =0, 

in which the co-efficients of y are entire, and that of the first 
term is 1. 

2d. When the denominators contain common factors, we shall 
evidently render the co-efficients entire, by making k equal to the 
least common multiple of all the denominators. But we can 
simplify still more, by giving to k such a value that k^^ F, k^^ . , . 
shall contain the prime factors which compose ^, c?, /, A, raised 
to powers at least equal to those which are found in the de- 
nominators. 

Thus, the equation 






becomes 



^ 6 ^ "^ 12 ^ 150^ 9000" ' 

y 

ttfter making x = -y-, and reducing the terms. 

First, if we make k — 9000, which is a multiple of all the 
other denominators, it is clear that the co-efficients become entire 
numbers. 

But if we decom.pose G, 12, 150, and 9000, into their prime 
factors, we find, 

6^2x3, 12 = 22x3, 150^:2x3x52, 9000 = 2^ < 32 x 5- 

and by making 

A ^ 2 X 3 X 5, 



828 ELEMENTS OF ALGEBRA. fCHAP. X 

the product of the different prime factors, we cbtain 

F = 22 X 32 X 52, /.-s ^ 03 X 33 X 53, Jc^ = 2* X 3* X 5* ; 
wkence we see that the values of k^ k"^^ A;^, A:*, contain the 
prime factors of 2, 3, 5, raised to powers at least equal to 
those which enter into 6, 12, 150, and 9000. Hence, making 

A- r= 2 X 3 X 5, 
is sufficient to make the denominators disappear. Substituting 
this value, the equation becomes 

_ 5.2.3.5 3 5.22.32.52 2 _ 7.23.33.53 _ 13.2*.34.5^ _ 
^ 2.3 ^ "* 2273"" ^ 2:3^52" ^ 23.32.53 ~ ' 

which reduces to 

y* — 5.5?/3 + 5.3.52y2 _ 7.22.32.5?/ - 13.2.32.5 = ; 
or, y^ — 25y3 + 37.5^2 _ lo^Oy - 1 170 =: 0. 

Hence, we perceive the necessity of taking k as small a 
number as possible : otherwise, we should obtain a transformed 
equation, having its co-efficients very great, as may be seen by 
reducing the transformed equation resulting from the supposi- 
tion k = 9000. 

Havmg solved the transformed equation, and found the values 
of y, the corresponding values of x may be found from *.he 

y 

equation, a: = — -, 

A.' 

by substituting for y and k their proper values. 

EXAMPLES. 

1- *'-^^' + ^-^-^ = " 

V 
Making x z=— ^ and we have, 

y3 _ X4^2 J^Uy — lb — 0. 

13 , 21 , 32 ,43 1 ^ 

2- ^-12^ + 40^^- 225^^- 600^- 800 = ''- 

Making x = -^^- = ^, and we have, 

y3 — 65y* 4- 1890^3 - 30T20y2 _ 928S00y - 972000 = 0. 



CHAP. X.] 



TRANSFORMATION OF EQUATIONS. 



829 



SecoTid. 

To make the second or any other term dlsappexr from an 
equation. 

263. The difficulty of solving an equation generally diminishes 
with the number of terms involving the unknown quantity. 

Thus the equation 

^^ — 1-> gives immediately, a; = ± .^/^ 
while the complete equation 

x"^ + 2px -{- q = 0, 
requires preparation before it can be solved. 

Now, any given equation can always be transformed into an 
incomplete equation, in which the second term shall be wanting. 

For, let there be the general equation, 

a^m _f_ p^.m-1 _j_ ^^«-2 _|. _ , -^ Tx-\- U=: 0. 

Suppose X = u -\- x^, 

u being a new unlmown quantity, and x' entirely arbitrary. 

By substituting if-{- x' for x^ we obtain 
(w -f ar")"* +P(w + a/)'^i + ^ (w + x'Y-'^ ...-k-T{u-\-x')^U= 0. 

Developing by the binomial formula, and arranging with refer, 
ence to w, we have 

1 






4- (m — 1) Px' 



+ . . . 



}^=:0. 



+ Tx' 

Since xf is entirely arbitrary, we may dispose of it in such 
wav that we shall have 



mtf -|- P = : whence, a/ == — . 



8S0 ELEMENTS OF ALGEBRA. [CHAP X. 

Substituting this value of x' in the last equation, we shall 
obtain an incomplete equation of the form, 

w"* + Q'W^^ + i^^W^-a -i- . . . T'u+ U' = 0, 

in which the second term is wanting. 

If this equation were solved, we could obtain any value of 
X corresponding to that of u, from the equation 

X = u -\- x\ snice x = u . 

m 

We have, then, in order to make the second term of an 
equation disappear, the following 

RULE. 

Substitute for the nnhiown quantity a new unknown quantity 
minus the co-eficient of the second term divided by the exponent 
which exjjr esses the degree of the equation. 

Let us apply this rule to the equation, 

x"^ + 2j:)X =z q. 
If we make x = u — p, 

we have (u — pY + 2p (u — p) = q* 

and by performing the indicated operations and transposing, 
we find 

^2 _ 2)2 _|_ g^ 

263*« Instead of making the second term disappear, it may 
be required to find an equation which shall be deprived of its 
third, fourth, or any other term. This is done, by making the 
co-efficient of u, corresponding to that term, equal to 0. 

For example, to make the third term disappear, we make, 
in the transformed equation, (Art. 263), 

m ^-i x'^ + {m - 1) Py + Q = 0, 

from which we obtain two values for x\ which substituted in 
the transformed equation, reduce it to the form, 

ym _|_ p'ym-\ _|_ R'u^-^ . . . 4- r^W -f- CT"' = 0. 



CHAP. X.] OF DERIVED POLYNOMIALS. 331 

Beyond the third term it will be necessary to solve an 
equation of a degree superior to the second, to obtain the value 
of x' ; and to cause the last term to disappear, it will be neces- 
sary to solve the equation, 

x'^ + Po;^"'^! . , , -^Tx' -^ TJ =% 
which is what the given equation becomes when x' is sub- 
stituted for X. 

It may happen that the value, 

m 
which makes the second term disappear, causes also the disap 
pearance of the third or some other term.- For example, in 
order that the third term may disappear at the same time 
with the second, it is only necessary that the value of xf^ 
which results from the equation, 

P 



shall also satisfy the equation, 
2 



m ^— x'"^ + (m - 1) Pa:^ + § = 0. 



P 

Now, if in this last equation, we replace x' by —■ — , we have 

m — 1 P2 pi 

[rn-\)—^q=.^^ or (m - 1) P2 _ 2^^ = ; 



2 iri? m 

and, consequently, if 



p2^ 2m^ 



m — 1' 

the disappearance of the second term will also involve that of 
the third. 

Formation of Derived Polynomials, 

264« Tliat transformation of an equation which consists in 
substituting u -\- x^ for x^ is of frequent use in the discussion 
of equations. In practice, there is a very simple method of 
obtaining the transformed equation which results from this sub 
s'ltutior^ 



332 ELEMENTS OF ALGEBRA. LCHAP. X 

To show this, let us substitute for x^ u -\- x' in the equation 

then, by developing, and arranging the te: tns according . to the 
ascending powers of u^ we have 



x"^ 


+ ma;''"-i 


J^Px^m~l 


+ (m-l)Px^'"-2 


+ ^^"-2 


+ (m-2)$a/'"'-3 


4- . . . 


+ . . . 


-{-Tx' 


+ r 


+ u 





u-\-m 



1 



1.2 



j^{jn~\)^^Px'-^ 
+ {m-2f-^Qx^^ 



u2+. 



^=0. 



By examining and comparhig the co-ef^cients of the ditlerent 
powers of v, we see that the co-efficient of u^, is what the first 
member of the given equation becomes when a/ is substituted 
in place of x ; we shall denote this expression by X\ 

The co-efficient of u^ is formed from the preceding term X', 
by multiplying each term of X'' by the exponent of a/ in that 
term, and then diminishing this exponent by 1 ; we shall denote 
this co-efficient by Y\ 

The co-efficient of v^ is formed from Y\ by multiplying each 
term of !P by the exponent of x^ in that term, dividing the 
product by 2, and then diminishing each exponent by 1. Repre- 

senting this co-efficient by — , we see that Z' is formed from V^ 

in the same manner that Y is formed from X\ 

In general, the co-efficient of any power of w, in the above 
transformed equation, may be found from the preceding co-efficient 
in the following manner, viz. : — 

Multiply each term of the preceding co-efficient hy the exponent 
of xf in that term^ and diminish the exponent of xf hy \ ; then 
divide the algebraic sum of *hese expressions by the number of pre- 
ceding co-efficients. 



CH-AJ*. X.J OF DERIVED POLYNOMIALS. 333 

The law by which the co-efficients, 



■^5 ^ ? 1 o 



Z' F 



1.2' 1.2.3' 

are derived from each other, is evidently the same as that 
which governs the formation of the numerical co-efficients of 
the terms in the binomial formula. 

The expressions, Y"^ Z\ P, IP, &c., are called successive de- 
rived polynomials of X\ because each is derived from the pre- 
ceding one by the same law that*!^'' is derived from X\ 

Generally, any polynomial which is derived from another by 
the law just explained, is called a derived 'polynomial. 

Recollect that X' is what the given polynomial becomes when 
a/ is substituted for x. 

Y' is called the first-derived polynomial ; 
Z^ is called the second- derived polynomial ; 
V is called the third-derived polynomial ; 
&c., &c. 

We should also remember that, if we make w = 0, we shall 
have oi/ — x, whence X^ will become the given polynomial, from 
which the derived polynomials will then be obtained. 

265. Let us now apply the above principles in the following 

EXAMPLES. 

1. Let it be required to find the derived polynomials of the 
first member of the equation 

Sx^ + 6x-3 — Sx^-{-2x + 1=0. 

Now, u being zero, and x' = x, we have from the law of 
form -J g ^he derived polynomials, 

X' = Sx^ + 6x^ — 3a;2 -|- 2.r + 1 ; 
Y' = 12:c3 -f ISu)'- -Qx -\-2', 
Z' - 36a;2 ^36^. _ e ; 
F' = 72ar -f 36; 
W = 72. 



334 



:lements 



It should be remarked that the exponent of a:, in the terms 1, 2, 
•— 6, 36, and 72, is equal to ; hence, each of those terms 
disappears in the following derived polynomial. 

2. Let it be required to cause the second term to disappear 
in the equation 

x^ — 122-3 4. 17^.2 _ 9^ _^ 7 ^ 0. 

12 
Make (Art. 263), x = u-{-—-:=u-\-S; 

whence, a/ = 3. 

The transformed equation will be of the form 

X^ + Tu + ^^2 + 2^^' + ^* = 0, 

and the operation is reduced to finding the values of the eo- 
efficients 

Y' W — 

"^' ' 2' 2.3* 

Now, it follows from the preceding law, for derived poly- 
nomials, that 
X' = (3)4-12.(3)3+17.(3)2-9.(3)^+7, or X^ =-110 



Y' = - 12^ 



=6.(3)2-36.(3)1 + 17, or 
= 4.(3)1-12 



_ =-37 



2.3 



= 0. 



Y' =4.(3)3-36.(3)2+34.(3)1-9, or 

2 

2T3 

Therefore, the transformed equation becomes 
^4 __ 37^^2 _ YlZu — 110 = 0. 

3. Transform ihe equation 

4^3 _ 5.^2 4. 7^^ _ 9 ^ 

Into another equation, the roots of which shall exceed those of 
the given equation by 1. 

Make, x ^zu — \\ whence a/ = — 1 : 

and the tiansformed equation will be of the form 



CHAP. X.J DERIVED POLYNOMIALS. 835 

. We have, from the principles established, 
X = 4.{-iy- 5.(- l)2 + 7.(-l)i-9, or X' = -25; 
F' = 12.(-1)2- 10.(- 1)1 +7 - - F'=+29; 

^=12. (-1)1-5 |- =-17; 

-— =4 - - -— = 4-4. 

2.3 2.3 ^ 

Therefore, the transformed equation is, 

4^3 — 17w2 + 292^ — 25 = 0. 

4. What is the transformed equation, if the second term be 
made to disappear from the equation 

x^ - 10^4 4- 7a;3 + 4a: - 9 = ? 

Ans. u^ — ZSu^-USu^-152u-7S = 0. 

5. What is the transformed equation, if the second term be 
made to disappear from the equation 

3a;3 + 15a:2 -f 25a: - 3 = 0? 

^ , 152 ^ 

27 

6. Transform the- equation 

3a:* - 13a:3 _j_ 7a;2 _ g^r _ 9 = 
into another, the roots of which shall be less than the roots of 

. . 1 

the given equation by — . 

o 

Ans. 3w* — 9zi3 — 4^^ ^ u = 0. 

9 3 

Properties of Derived Polynomials. 

266i We will now develop some of the properties of derived 
polynomials. 

Let x"^ + Px^-^ + (2a:"^2 , , , Tx -^ XJ = ^ 

be a given equation, and a, 6, c, c?, &;c., its m roots. We shall 

then have (Art. 248), 

r« + Pa;'»-i 4- ^a;'^^ . . . = (a; - a) (r — ^) (a; — c) . . . (a; — /), 



336 ELEMENTS OF ALGEBRA. [CHAP. X 

Making x = s/ -}- u^ 

or omitting the accents, and substituting x -\- u for x, and we have 
{x + w)'» -f F{x + w)"*-^ + . . . ={x-{- u — a)(x-\-u — b) . . ,] 
or, changing the order of x and w, in the second member, and 
regarding x — a, x — b, . . . each as a single quantity, 



{x 4- w)*" +P(:c 4- w)"*-^ ... — (w -f ic —a) {u+x—b) . . . (w + ^— /). 

Now, by performing the operations indicated in the two 

members, we shall, by the preceding article, obtain for the first 

member, 

X-\- Tu-^—-u'^+ . . . w"»; 
(^ 

X being the first member of the proposed equation, and Y, Z, &c., 

the derived polynomials of this member. 

With respect to the second member, it follows from Art. 251 : 
1st. That the term involving u^, or the last term, is equal to 

the product {x — a) (x — b) . . . (^x — I) of the factors of the 

proposed equation. 

2d. The co-effiGient of u is equal to the sum of the products 
of these m factors, taken m — 1 and m — 1. 

3d. The co-efficient of u"^ is equal to the sum of the products 
of these m factors, taken m — 2 and m — 2 ; and so on. 

Moreover, since the two members of the last equation are 
identical, the co-eflicients of the same powers of u in the two 
members are equal. Hence, 

X={x-a)(x-b){x-c) . . . {x- Z), 
which was already shown. 

Hence, also, Y, or the first derived polynomial, is equal to the 
sufn of the products of the m factors of the first degree in the pro- 
posed equation^ taken m — 1 and m — 1 ; or equal to the algebraic 
sum of all the quotients that can be obtained by dividing X by 
each of the m. factors of the first degree in the proposed equation ^ 
iJiai is, 



X — a x — b X — c ' X — r 



CHAP X.] EQUAL ROOTS. 837 

Also, — , that is, the second derived polynomial, divided by 2, 

is equal to the sum of the products of the m factors of the Jint 
member of the proposed equation, taken m — 2 and m— 2/ or 
equal to the sum of the quotients obtained by dividing X by each 
of tht, different factors of the second degree ; that is, 

Z X X X 

+ 



2 {x ~ a) [x — b) (x — a) [x — c) (x — k){x — I) 

and £0 on. 

0/ Equal Boots. 

267, An equation is said to contain equal roots, when its first 
member contains equal factors of the first degree with respect to 
the unknown quantity. When this is the case, the derived poly- 
nomial, which is the sum of the products of the m factors taken 
m — 1 and m — 1, contains a factor in its different parts, which 
is two or more times a factor of the first member of the pro- 
posed equation (Art. 266) : hence, 

There must be a common divisor between the first member of the 
proposed equation, and its first derived polynomial. 

It remains to ascertain the relation between this common divi- 
sor and the equal factors. 

268* Ho.ving given an equation, it is required to discover whether- 
it has equal roots, and to determine these roots if possible. 

Let us make 

X= x"^ -j- Fx"^- -f Qx"^-^ -i- . . . + Tx-^- U=0, 
and suppose that the second member contains n factors equal to 
sc — a, n' factors equal to x — b, n" factors equal to a; — c . . ., 
and also, the simple factors x —p, x — q, a; — r . . . ; we sliali 
tlien have, 
X .-{x- ay {x - by [x — cy , , , (x-p){x-q)(x-r) (1). 

We have seen that Y, or the derived polynomial of X, is 
tied sum of the quotients obtained by dividing X by each of the m 
factors of the first degree in the proposed equation (Art. 266). 

22 



338 ELEMENTS OF ALGI^ERA. [CHAP. X. 

Now, since X contains n factors eqaal to a; — «, we shall 

X 
ha^'e ?i partial quotients equal to • ; and the same Treason 

ing applies to each of the repeated factors, x — 6, x -- c 

Moreover, we can form but one quotient for each simple flictor, 
which is of the form, 

X X X 

X — p* X — ^' X — r ' ' ' ' 

therefore, the first derived polynomial is of the form, 

X — a X — X — c X — p X — q x — r 

By examining the form of the value of X in equation (J), 
it is plain that 

{x ~ «)«-!, {x - hy-\ {x - c)«"-i . . . 
are factors common to all the terms of the polynomial Y; 
hence the product, 

{x ~ a)"-i X {x — ly-^ X {x — c)""-i . . . 
is a divisor of Y. Moreover, it is evident that it will . alsc 
divide X: it is therefore a common divisor of X and Y-, and 
it is their greatest common divisor. 

For, the prime factors of X, are x —a, x — b^ x —c . . ., and 

X —p, X — q, X — r . . .; now, x — p, x — q, x — r, cannot 

divide Y, since some one of them will be wanting in some of 

the parts of Y, while it mil be a factor of all the other parts. 

Hence, the greatest common divisor of X and Y, is 

D z=(x — a)"-i (x — by-^ {x — cY"-^ . . . ; that is. 

The greatest common divisor is comioosed of the product of those 
factors which enter tivo or more times in the given equation, each 
raised to a power less by 1 than in the primitive equation. 

269. From the above, we deduce the following method fiji 
finding the equal roots. 

To discover whether an equation, 

X=0, 
contains any equal roots : 



i 



CHAP. IX.J EQUAL ROOTS. 3o9 

1st. Form Y^ or the derived pohjnomial of X i then seek for 
the greatest common divisor between X and Y, 

2d. If one cannot he obtained^ the equation has no equal roots, 
or equal factors. 

If we find a common divisor D, and it is of the first degree, 
or of the form x — h, make x — h = 0, whence x — h. 

We then conclude, that the equation has two roots equal to h, 
and has but one species of equal roots, from which it may be 
freed by dividing X by (x — h)^. 

If D is of the second degree with reference to ar, solve thi 
equation D — 0. There may be two cases ; the two roots will 
be equal, or they will be unequal. 

1st. When we find D =z [x — h)^, the equation has three roots 
equal to h, and has but one species of equal roots, from which 
it can be freed by dividing X by (x — h)^. 

2d. When D is of the form {x — h){x — A'), the proposed 
equation has tioo roots equal to h, and two equal to h', from 
which it may be freed by dividing X by {x — hy {x — h'Y, 
or by D^. 

Suppose now that D is of any degree whatever ; it is necessary, 
in Older to know the species of equal roots, and the number 
of roots of each species, to solve completely the equation, 

J) = 0, 

Then, every simple root of the equation D =1 will be twice a 
root of /he given equation; every double root of the equation D z=z 
will be three times a root of the given equation * and so on.. 

As to the simple roots of 

X=0, 

we begin by freeing this equation of the equal factors contained 
in it, and the resulting equation, X' = 0, will make known the 
simple roots. 



340 ELEMENTS OF ALGEBKA. [CHA.P. X. 

EXAMPLES. 

1. Determine whether the equation, 

2x* — 12^3 + 19a:2 — 6:r + 9 = 0, 

contains equal roots. 

We have for the first derived polynomial, » 

8:^3 — 36a;2 + 2Sx — 6. 

Now, seeking for the greatest common divisor of these poly 
nomials, we find 

D = X — S = 0, whence x = S: 
hence, the given equation has two roots equal to 3. 
Dividing its first member by (x — 3)^, we obtain 

2a;2 + 1 — ; whence, x = ± — V— 2. 

The equation, therefore, is completely solved, and its roots are 

2. For a second example, take 

x5 — 2x^ + Sx^ — 7x^-\-Sx — S = 0. 
The first derived polynomial is 

5^4 _ 8a;3 -f 9a;2 - 14a; 4- 8; 
and the common divisor, 

x^-2x + I = {x-^IY: 

hence, the proposed equation has three roots equal to 1. 
Dividing its first member by 

{x - 1)3 = x^ - Sx^ + 3rr - 1, 
the quotient is 



a:2 -f- a; 4- 3 ^ ; whence, x ^ ^ 



- 1 ± v~ n 

2 

thus, the equation is completely solved. 



CHAP. X.J EQUAL ROOTS. 841 

3. For a third example, tak^ the equation 

^.7 + 5^.6 4. G_^.5 _ (3.^4 ^. 15^.3 _ 3^.2 _|_ 82: -h 4 = 0. 

The first derived polynomial is 

Ix^ + 30a:5 + 3O2;* - 24a:3 _ ^bx^ - 6a; + 8 ; 
and the common divisor is 

x^ 4- ox^ -[- x"^ — Zx — 2. 
The equation, 

a;4 _|_ 3^3 _j_ 3.2 _ 3^ _ 2 = 0, 
cannot be solved directly, but by applying the method of equal 
roots to it, that is, by seeking for a common divisor between 
its first member and its derived polynomial, 

4a;3 + ^x^ + 2a^ — 3 : 
we find a common divisor, a; + 1 ; which proves that the square 
of a; + 1 is a factor of 

x^ + 3a;3 + a;2 — 3a; — 2, 
and the cube of a; + Ij a factor of the first member of the 
given equation. 
Dividing 

x^ + 3:^3 _j_ ^2 _ 3^ _ 2 by {x + 1)2 =:x^-\-2x-{-\\ 

we have x'^ -{- x —2, which being placed equal to zero, gives 
the two roots, a; = 1, a; = — 2, or the two factors, x — 1 and 
X -\-2. Hence, we have 

X* + ox^ H- a;2 — 3a; - 2 = (a; + 1)2 {x — 1) {x + 2). 
Therefore, the first member of the proposed equation is equal to 
(a; + 1)3(0; -1)2 (a; + 2)2; 
that is, the proposed equation has three roots equal to — 1, two 
equal to +1, and two equal to — 2. 

4. What is the --product of the equal factors of the equation 
x' — 7x^ + 10a;5 + 22a;* - 43a;3 - 35a;2 + 48a; + 36 = ? 

Ans. (a; -2)2 (a; -3)2 (a; + 1)'. 

5. What is the product of the equal factors in the equatioii, 

a:7 _ 3.6 + 9^5 _ 19^4 ^ 27a;3 - 33a;2 + 27a; - 9 == ? 

Ans. (a;-l)3(a;2 + 3)2. 



\ 

342 ELEMENTS OF ALGEBRA. [CHAP. I. 

Elimination. 

270* We have already explained the methods of eliminating 
oue iinkno^Ti quantity from two equations, when these equations 
are of the first degree with respect to the unknown quantities. 

When the equations are of a higher degree than the first, 
the methods explained are not in general applicable. In this 
case, the method of the greatest common divisor is considered the 
best, and it is this method that we now propose to investigate. 

One quantity is said to be a function of another when it de- 
pends upon that other for its value ; that is, when the quan- 
tities are so connected, that the value of the latter cannot be 
changed without producing a corresponding change in the former. 

271. If two equations, containing two unkno^vn quantities, be 
combined, so as to produce a single equation containing but one 
unknown quantity, the resulting equation is called a final equa- 
tion ; and the roots of this equation are called comj^atihle 
values of the unkno\vn quantity which enters it. 

Let us assume the equations, 

P = and ^ = 0, 

in which P and Q are functions of x and y of any degree 
whatever ; it is required to combine these equations m such a 
manner as to eliminate one of the unkno^\^l quantities. 

If we suppose the final equation involving y to be found, and 
that 2/ = a is a root of this equation, it is plain that this value 
of y, in connection with some value of x^ will satisfy both 
equations. 

If then, we substitute this value of y in both equations, there 
Mill result two equations containing only x, ^id these equations 
will have at least one root in common, and conseqiiently, their 
first members will have a common divisor involving x (Art. 246). 

This common divisor will be of the first, or of a higlier degree 
with respect to a:, according as the particular value of y = tx cor 
responds to one or more values of x. 



CHAP. XI. J ELIMINATION. 843 

Conversely, every value of y which, being substituted in the 
two equations, gives a common divisor involving x, is necessarily 
a comjiatible value, for it then satisfies the two equations at the 
same time with the value or values of x found from this common 
divisor when put equal to 0. 

272. We will remark, that, before the substitution, the first 
members of the equations cannot, in general, have a common divi- 
sor which is a function of one or both of the unknown quantities. 

For, let us suppose, for a moment, that the equations 

P = and ^ = 0, 
are of the form 

F' X B = and Q' X R = 0, 
R being a function of both x and y. 

Placing It =z 0, we obtain a single equation involving two 
unknown quantities, which can be satisfied with an infinite number 
of systems of values. Moreover, every system which renders R 
equal to 0, would at the same time cause F' . R and Q'.R to 
become 0, and consequently, would satisfy the equations 
P = and Q:= 0. 

Thus, the hypothesis of a common divisor of the two poly, 
nomials F and Q, containing x and y, brings with it, as a con- 
sequence, that the proposed equations are indeterminate. There- 
fore, if there exists a common divisor, involving x and y, of the 
two polynomials F and Q, the proposed equations will be inde- 
terminate, that is, they may be satisfied by an infinite number 
of systems of values of x and ?/. Then there is no data to 
determine a final equation in y, since the number of values of y 
is i) I finite. 

Again, let us suppose that P is a function of x only. 

Placing R =z a, we shall, if the equation be solved willj 
reference to x, obtain one or more values for this unI;nown 
quantity. 

Each of these values, substituted in the equations 
F' .R^O and Q\ R =: 0, 



844 ELEMENTS OF ALGEBRA. [CHAP. X 

will satisfy them, whatever value we may attribite to y, smco 
these values of x would reduce R to 0, independently of y. 
Therefore, in this case, the proposed equations admit of a finite 
number of values for x, but of an infinite number of values for 
y and then, therefore, there cannot exist a final equation in y. 
Hence, when the equations 

are determinate, that is, when they admit only of a limiied 
number of systems of values for x and ?/, their first members 
cannot have for a common diviaor a function of these unknown 
quantities^ unless a particular substitution has been made for one 
of these quantities. 

273i From this it is easy to deduce a process for obtaining 
the final equation involving y. 

Since the characteristic property of every compatible value 
of y is, that bemg substituted in the first members of the two 
equations, it gives them a common divisor involving a*, which 
they had not before, it follows, that if to the two proposed 
polynomials, arranged with reference to a?, we apply the process 
for finding the greatest common divisor, we shall generally not 
find one. But, by continuing the operation properly, we shall 
arrive at a remainder independent of a:, but which is a function 
of y, and which, placed equal to 0, will give the required final 
equation. 

For, every value of y fnmd from this equation, reduces to 
zero the last remainder in the operation for finding the common 
divisor ; it is. then, such that being substituted in the preceding 
remainder, it will render this remainder a common divisor of the 
first members P and Q. Therefore, each of the roots of the 
equation thus formed, is a compatible value of y. 

274i Admitting that the final equation may be completely 
solved, which would give all the compatible values, it would 
afterward be necessary to obtain the corresponding values of x. 
Now, it is evident that it would be sufficient for this, to sub- 
stitute the different values of y in the remainder preceding the 



CHAP. X.] JiJLIMINATION-. 845 

last, put the polynomial involving x which results from it, equal 
to 0, and find from it the values of .r; for these polynomials 
are nothing more than the divisors involving a;, which become 
common to A and B. 

But as the final equation is generally of a degree superior to 
llie second, W€ cannot here explain the methods of finding the 
values of y. Indeed, our design was principally to show that, 
two equations of any degree being given, we can, without supposing 
the resolution of any equation^ arrive at another equation, contain- 
ing only one of the unknown quantities which enter into the pro* 
posed equations. 

EXAMPLES. 

1. Having given the equations 

a;2 -j- ary + y2 _ 1 = Q, 
x^-^y'^ = 0, 
to find the final equation in y. 



First Op 
x^ + 2/3 


oration. 
\x^-\-xy~\-y^- 


-1 


.^3 4. y.^.2 _|. (y2 _ 1)^ 


X —y 




— yx"^ — {y^ — I) X -\- y^ 

— yx"^ — y'^x — y^ -\- y 







X + 2y^ — y = 1st remainder. 
Second Operation. 

x^ -\- yx -\- y^ — 1 \\x -f- 2y3 _ y 

x^-^(2y^-y)x 1 ^ _ (2^/3 _ g^,) 

- (2y' - 2y) a: - 4y^ + 6y^ - 2y^ 
4/ - 6y* + 3y2 _ 1. 
Hence, the final equation in y, is 

4y^-Gy' + Sy^—l=0. 



346 ELEMENTS OF ALGEBRA. [CILA P. X 

If it were required to fjud the final equation in «:, we observe 
that X and y enter into the primitive equations under the same 
forms ; hence, x may be changed into y and y into x^ without 
destroying the equality of the members. Therefore, 

' 4:r6 — Qx^ + Zx"^ — 1 =0 
is the final equation in x. 

2. Find the final equation in y, from the equations 
x\-^yx'^ + {^y'^-y-\-l)x-y^-\-y'^-2y^(}, 
x^ — 2yx+ ^f — y — 0. 



First Operation, 
x^ — ^yx^ + (?/2 _ y-) X 



^ - Zyx-^ + (3y2 - y j^ i)x - y^ + y"^ - ^y Vx"^ - 2xy + y"^ - y 



- yx^ + (2y2 J^l)x-y^^y'^-2y 




~ yx^+ 2y'^x - y' + y^ 




x-2y 




Second Operation. 




x^ — 2xy -\. y"^ — y\ X- 


-2y 


x2 _ 2xy 


X 




y^ -y- 






Hence, 2/2 — y — 0, 







is the final equation in y. This equation gives 

y = I and y = 0. 
Placing the preceding remainder equal to zero, and substi- 
tuting therein l^e values of y, 

y = 1 and y = 0, 
we fmd for the iorresponding values of x, 

X z=i 2 and x — ; 
from which <^r ^iven eqrations may be entirely solved. 



CHAPTER XI. 

SOLUTION OF NJMER IJAL EQUATIONS CONTAINING BUT ONE UNKNOWN 

QUANTITY. — Sturm's theorem. — cardan's rule. — horner's method. 

275. The principles established in the precedhig chapter, are 
applicable to all equations, whether the co-efficients are numerical 
or algebraic. These principles are the elements which are em- 
ployed in the solution of all equations of higher degrees. 

Algebraists have hitherto been unable to solve equations of a 
higher degree than the fourth. The formulas which have beer 
deduced for the solution of algebraic equations of the higher 
degrees, are so complicated and inconvenient, even when they 
can be applied, that we may regard the general solution of an 
algebraic equation, of any degree whatever, as a problem more 
curious than useful. 

Methods have, however, been found for determining, to any 
degree of exactness, the values of the roots of all numerical 
equations ; that is, of those equations which, besides the unknown 
quantity, involve only numbers. 

It is proposed to develop these methods in this chapter. 

276* To render the reasoning general, we will take the 
equation, 

X:=z x"^ -{- Fx"^^ + g.r'"-2 4- . . . U= 0. 
hi which P, Q . . . denote particular numbers which are real, 
and either positive or negative. 

]^ we substitute for x a number a, and denote by A what 
X becomes under this suppositi )n ; and again substitute a -{- u 
foi X. and denote the^new polynomial by A^ : then, u may he 
taken rt small, that the differehce between A' and A shall bt 
less than any assignable quantity. 



318 ELEMENTS OF ALGEBRA. [CHAP. XL 

If, no^v, we denote hj B, C, D, . . . . what the co-efficients 

Z V 
F, — , - — - (Art. 264), become, wh^ri we make a: = a, we 
Z jZ . 3 

shall have, 

A' = A'i-Bu+ Cu^ + Du^ ^ . . . ^iT - - . (1) ; 
whence, 

A' -'A = Bu-\- Cu'^-\-Du^-\- . . . -\-u'^ - - - (2). 
It is now required to show that this difference may be ren- 
dered less than any assignable quantity, by attributing a value 
sufficiently small to u. 

If it be required to make the diffi3rence between A^ and A 
less than the number li, we must assign a value to u which 
will satisfy the inequality 

Bu + Cu^ + Bu^ + w"* < A" - - - (3). 

Let us take the most unfavorable case that can occur, viz., 
let us suppose that every co-efficient is positive, and thaf each 
is equal to the largest, which we will designate by IT. Then 
any value of u which will satisfy the inequality 

K{u + w2 4. ^3 _|_ ^^m^ ^js/- ... ^4j^ 

will evidently satisfy inequality (3). 

Now, the expression within the parenthesis is a geometrical 
progression, whose first term is u, whose last term is w'", and 
whose ratio is u ; hence (Art. 188), 

u+ u^+u^-\- . . .u"^ = = =- X (1 — «*). 

u — 1 I — u I — u ^ ' 

Substituting this value in inequality (4), we have, 

Ku 



1 -u 



(1 - w-)< iY - - - - (5). 



N 
Vi now we make u = -— , the first factor of the nrst mem 

JS \- A 

(N \^ 
~v — in ^^ ^®^' 

than 1, the second factor is less than 1; hence, the fiist mem 
ber is less than N, 



CHAP. JCl.] NUMERICAL EQUATIONS. 849 

N 
We conclude, therefore, that u = -— -• and every smaller 

value of w, will satisfy the inequalities (3) and (4), and conse- 
quently, make the difference between A' and A less than any 
assignable number JSf. 

If in the value of A\ equation (1), we make u= — , it 

is plain that the sum of the terms 

Bu 4- Cu^ + Du^ + . . . -w" 
will be less than A^ from what has just been proved ; whence 
we conclude that 

In a series of terms arranged according to the ascending powers 
of an arhitrarij quantity^ a value may he assigned to that 
so small, as to make the first term numerically greater than the 
sum of all the other terms. 

First Principle, 

277 • If two numbers p and q, substituted in succession in the 
place of X in the first member of a numerical equation, give results 
affected with contrary signs, the proposed equation has a rettl root, 
comprehended between these tivo numbers. 

Let us suppose that p, when substituted for x in the first 
member of the equation 

X = 0, gives -h B, 
and that q, substituted in the first member of the equation 
X = 0, gives —E\ 

Let us now suppose x to vary between the values of p and q 
by so small a quantity, that the difference between any two 
corresponding consecutive values of X shall be less than any 
assignable quantity (Art. 276), in which case, we say that X is 
subject to the law of continuity, or that it passes through all 
the intermediate values between B and — B'. 

Now, a quantity w^hich is constantly finite, and subject to the 
'aw of continuity, cannot change its sign frcm positive to ne^a 



c 

850 ELEMENTS OF ALGEBRA. [CHAP. XI. 

tive, or from negative to positive, without passing through zero : 

hence, there is at least one number between 'p and q which will 

satisfv the equation 

X=0, 

and consequently, one root of the equation lies between these 
numbers. 

278. We have shown in the last article, that if two numbers 
be substituted, in succession, for the unknown quantity in any 
equation, and give results affected with contrary signs, that there 
will be at least one real root comprehended between them. We 
are not, however, to conclude that there may not be more ^han 
one ; nor are we to infer the converse of the proposition, "snz., 
that the substitution, in succession, of two numbers which include 
roots of the equation, will necessarily give results affected with 
contrary signs. 

Second Principle. 

279. When an uneven numher of the real roots of an equation 
is comprehended between two numbers^ the results obtained by sub- 
stituting these numbers in succession for x in the first member, will 
have c&ntrary signs; but if they comprehend an even number of 
roots, the results obtained by their substitution will have the same sign. 

To make this proposition as clear as possible, denote by 
a. b, c, . . . those roots of the proposed equation, 

which are supposed to be comprehended between p and q, and 
by F", the product of the factors of the first degree, with refer- 
ence to X, corresponding to the remaining roots of the given 
equation. 

The first member, X, can then be put under the form, 

{x-a){x-b)(x-c) ... X F=0. 
Now, substituting p and q in place of z, in the first meiO" 
Der, ve shall obtain the two results, 

(^p-a){p-b){p-c) ... X r, 
{q-<i){q-b){q-c) . . X Y". 



CHAP. XI.] NUMERICAL EQUATIONS. 351 

F"' and Y^^ representing what Y becomes, when we replace in 
Buccession, i hy p and q. These two quantities Y^ and JP^, are 
affected with the same sign; for, if they were not, by the first 
principle there would be at least one other real root com- 
prised between p and q, which is contrary to the hypothesis. 
To determine the signs of the above results more easily, 
divide the first by the second, and we obtain 

{p-o){p-b){p-c) . . . X Y' 
(g_a)(q-b){q-c) ... X Y^' 

which can be written thus, 

p — a p — b p — c Y' 

X J X X . . « ^T//' 

q — a q — b q — c Y^^ 

Now, since the root a is comprised between p and g, that 
is, is greater than one and less than the other, p — a and 
q — a must have contrary signs ; also, p — b and q — b must 
have contrary signs, and so on. 

HoLce, the quotients 

p — a p — b p — c 

^ , "^ 7, ^- , &C., 

q — a q — b q — c 
are all negative. 

Y' 

Moreover, -— ■ is essentially positive, since Y' and Y" are 

affected with the same sign ; therefore, the product 
p — a p — b p — c Y' 

X £ X X . . . ■xrri') 

q — a q — b q — c Y 

will be negative^ when the number of roots, a, 6, c . . ., com 
prehended between p and q^ is uneven, and positive when the 
number is even. 

Consequently, the two results, 

^p-a){p-b)(p -c) . . . X Y', 

End {q — ci){q — ^) {q — c) ' ' ' X Y'', 

will have contrary signs when the number of roots comprised 
between p and q is uneven^ and the same sign when the num- 
ber is even 



352 ELEMENTS OF ALGEBRA. [CHAP. XL 

Third Principle. 

280t If the signs of the alternate terms of an equation be 
changed, the signs of the roots will be changed. 

Take the equation, 

^m ^ p^m-l _|_ Q-^m-2 , _ ^ J^ _ Q . . (1) ; 

and hj changing the signs of the alternate terms, we have 

x'^ — Fx'^^ + Qx"^^ . . . dz U=0 - - (2), 
or, — a;"* + Px"^^ — ^^"^2 _ ^ ^ f^ _ . . (3). 

But equations (2) and (3) are the same, since the sum of the 
positive terms of the one is equal to the sum of the negative 
terms of the other, whatever be the value of x. 

Suppose a to be a root of equation (1) ; then, the substitution 
of a for X will verify that equation. But the substitution of 
— a for X, in either equations (2) or (3), will give the same 
result as the substitution of + a, in equation (1) : hence — a, 
is a root of equation (2), or of equation (3). 

We may also conclude, that if the signs of all the terms 
be changed, the signs of the roots will not be altered. 

Limits of Real Roots. 

281. The different methods for resolving numerical equations, 
consist, generally, in substituting particular numbers in the pro- 
posed equation, in order to discover if these numbers verify it, 
or whether there are roots comprised between them. But by 
reflecting a little on the composition of the first member of 
the general equation, 

x^ + Px^-^ + ^.r'"-^ ... -\- Tx-\- U^ 0, 

we become sensible, that there are certain numbers, above which 
it would be useless to substitute, because all numbers above a 
certain limit would give positive results. 



CHAP. XI.! LIMITS OF KEAL KOOTS. S53 

282. It is now reqiiired to determine a number^ which being 
substituted for x i?i the general equation^ will render the Jirst term 
.V™ greater than the a7'ithmetical sum of all the other terms ; 
that is, it is required to find a number for x which will render 

^m y p^m-l ^ Qj,m-2 J^ ^ ^ ^ J^ Tx + U, 

Let k denote the greatest numerical co-efficient, and substitute 
it in place of each of the co-efficients; the inequality will then 
become 

a:"' > kx"^^ -f kx"^-"^ -f ... -{- kx -{- k. 

It is evident that every number substituted for x which will 
satisfy this condition, will saLLfy tlio preceding one. Now,. 
dividing both members of this inequality by x^, it becomes 

k , k , k , , k k 

1 >— + — + — +••+ -^i^ + —n 

*l/ tJU U/ U/ Uf . 

Making x = k, the second member reduces to 1 plus the 
sum of several fractions. The number k will not therefore 
satisfy the inequality ; but if we make x = k -\- 1, we obtain 
for the second member the expression, 

K fC fC fC fC 



k-^i ' (k + iy ' (k-j- ly ' ' {k + i)*"-! ' (k + ly 

This is a geometrical progression, the first term of which is- 

k k 1 

the last term, 77— t-tt"' ^^^ *^® ratio, - — -— -; hence,. 



yt-f 1' ' (^+1)'"' ' A; + 1 

the expression reduces to 

k k 

1 J -^ {k^ir 



k-{- 1 

which is evidently less than 1. 

Now, any number > (^ + 1), put in place of a:, will render 

k k 

the sum of the fractions 1 -f- . . . still less : therefore, 

X x^ 

The greatest co-efficient phis 1, or amj greater number^ being 
substituted for x, will render the first term x™ greater than the 
arithmetical sum of all the other terms. 

23 



854 ELEMENTS OF ALGEBEA. [CHAP. XI. 

283. Every number which exceeds the greatest of the positive 
roots of an equation, is called a superior limit of the positive roots. 

From this definition, it follows, that this limit is susceptible 
of an infinite number of values. For, when a number is found 
to exceed the greatest positive root, every number greater than 
this, is also a superior limit. The term, however, is generally 
applied to that value nearest the value of the root. 

Since the greatest of the positive roots will, when substituted 
fur re, merely reduce the first member to zero, it fallows, that 
we shall be sure of obtaining a superior limit of the positive 
roots by finding a number^ which substituted in place of x, renders 
the first member positive, and which at the same time is such^ that 
every greater number will also give a positive result; hence, 

The greatest co-efficient of x plus 1, is a superior limit of 
the positive roots. 

Ordinary Limit of the Positive Roots. 

284, The limit of the positive roots obtained in the last article, 
is commonly much too great, because, in general, the equation 
contains several positive terms. We will, therefore, seek for a 
limit suitable to all equations. 

Let x^-^ denote that power of x that enters the first nega- 
tive term which follows a:"*, and let us consider the most unfavor- 
able case, viz., that in which all the succeeding terms are negative, 
and the co-efficient of each is equal to the greatest of ihe nega- 
tive co-efficients in the equation. 

Let S denote this co-efficient. What conditions will render 

a;"» > Sx"^-^ + Sx"^--^-^ -^ , . . Sx^- SI 
Dividing both members of this inequality by a:'", we hav«> 

x^ rC^+i a;''+2 + • • • + ^„^_i + ^^' 
Now, by supposing 

X = 'l/>S+ I, or for simplicity, making '!l/~S~z= S. 
which gives, S' = S"", and x = S' -h I, 



CHAP. XI.] LIMITS OF POSITIVE ROOTS. 865 

the second member of the inequality will become, 

+ • • • r / or , , \ , + 



wliich is a geometrical progression, of which is the 

(o -j- 1)" 

first term, and the ratio. Hence, the expression for the 

♦b + 1 

sum of all the terms is (Art. 188), 
S'^ ^S'^" 

1 _ ~ (S'-\-iy-' {S' -{- 1)'"^ 

Moreover, every number > 5^ + 1 or 'l/~^+ 1, will, when 
substituted for x, render the sum of the fractions 

etill smaller, since the numerators remain the same, while the 
denominators are increased. Hence, this sum will also be less. 



Hence, y ^ + 1, and every greater number, being substituted 
for X, will render the first term x^ greater than the arithmetical 
sum of all the negative terms of the equation, and will conse 
quently give a positive result for the first member. Therefore, 

That root of the numerical value of the greatest negative co-efi- 
cient whose index is equal to the number of terms which precede 
the first negative term^ increased by 1, is a superior limit of the 
'positive roots of the equation. If the co-efficient of a term is 0, 
Uie term must still be counted. 

Make n = 1, in which case the first negative term is the 
second term of the equation ; the limit becomes 

that is, the greatest negative co-efficient plus 1. 

Let 72 = 2 ; then, the limit is y^4- 1. When •» = 3. nhe 
limit is y^+ 1. 



556 ELEMENTS OF ALGEBEA. '"CHAP. XL 

EXAMPLES. 

1. What is the superior limit of the positive roots of rlie 
equation 

x^ - 5a;3 + 37a:2 _ Sa; + 39 = 0? 

Ans. "y^ + 1- = ]^ + 1 = 6. 

2. What is the superior limit of the positive roots of the 
equation 

a;5 _{. 7a;4 _ 12.^3 - 49^2 ^ 52a; - 13 = 1 

Ans. y^^- 1 =y^^+ 1 = 8. 

3. What is the superior limit of the positive roots of the 
equation 

x^-^ lla;2 — 25a; — 67 = 01 

In this example, we see that the second term is wanting, that 
is, its co-efficient is zero ; but the term must still be counted in 
fixing the value of n. We also see, that the largest negative 
co-efficient of x is found in the last term where the exponent of 
X is zero. Hence, 

\fS+ 1 .= 3^7 + 1 . 

and therefore, 6 is the least whole number that will certainly 
fulfil the conditions. 

Smallest Limit in Entire Numbers. 

285. In Art. 282, it was sho^^n that the greatest co efficient 
of X plus 1, is a superior limit of the positive roots. In the 
last article we found a limit still less ; and we now propose to 
'find the smallest limit, in whole numbers. 

Let X = 

be the proposed equation. If in this equation we make ar = a/ 4- « 
vf being arbitrary, we shall obtain (Art. 264), 



X'+F^2.+|t^2+ . . . +^^" = (1). 



I 



CHAP. XI.] LIMITS UF POSITIVE ROOTS. 357 

Let us suppose, that after successive trials we have determined 
a number for a/, which substituted in 

7/ 

renders, at the same time, all these co-eflicients positive, this nun* 
ber will in general be greater than the greatest positive root 
of the equation 

For, if the co efficients of equation (1) are all positive, no 
positive value of u can satisfy it ; therefore, all the real values 
of u must be negative. But from the equation 

X z= x' -{• u, we have u = x — x^ ; 
and in ordei that every value of u, corresponding to each of the 
values of x and a/, may be negative, it is necessary that th6 
greatest positive value of x should be less than the value of x'. 
Hence, this valuje of x' is a superior limit of the positive 
roots. If we now substitute in succession for x in X the values 
x' — \, x' — 2, x' — 3^ &c., until a value is found which will 
make X negative, tlien the last number which rendered it posi- 
tive will be the least superior limit of the positive roots in 
whole numbers. 

EXAMPLE. 

Let a;* - ^x^ - (jx"^ -1^x^1 = 0. 

As a/ is indeterminate, we may, to avoid the inconvenience 
of writing the primes, retain the letter x in the formation of 
the deri ved polynomials ; and we have, 

X = x^- 5^-3 - 6a:2 _ 19^ _|_ 7^ 
F = 4a;3 - 15a;2 _ 12:c -19, 

V 
^-=4. -5. 

The question is now reduced to fniding the smallest entire 
number which, substituted in placfe of x, will render all of 
these polynomials positive. 



358 . ELEMENTS OF ALGEBRA. LCHAP. XL 

It is pla'n that 2 and every number > 2, will render the 
poljTioniial of the first degree positive. 

But 2, substituted in the polynomial of the second degree, 
gives a negative result ; and 3, or any number > 3, gi N'es a 
positive result. 

Now, 3 and 4, substituted in succession in the polynomial 
of the third degree, give negativ« results ; but 5, and any 
greater number, gives a positive result. 

Lastly, 5 substituted in X, gives a negative result, and so 
does 6 ; for the first three terms, x^ — 5x^ — Qx'^, are equiva- 
lent to the expression x^ {x — 5) — Gx^, which reduces to when 
X = 6; but X =:7 evidently gives a positive result. Hence 7, is 
the least limit in entire numbers. We see that 7 is a supe- 
rior limit, and that 6 is not; hence, 7 is the least limit, as 
above shown. 

2. Applying this method to the equation, 

x^ — Sx^- Sx^ — 25a-2 _|_ 4;^ _ 39 _ 0, 
the superior limit is found to be 6. 

3. We find 7 to be the superior limit of the positive roots 
of the equation, 

a^5 _ 5,^4 _ 13.^.3 _|_ 17^2 _ 69 = 0. 

This method is seldom used, except in fuidlng incommen- 
surable roots. 

Superior Limit of Negative Boots. — Inferior Limit of Posi 
tive and Negative Roots. 

286. Having found the superior limit of the positive roots, 
it remains to find the inferior limit, and the superior and in- 
ferior limits of the negative roots, numerically considered. 

First^ If, in any equation, 

X = 0, we make x = — , 

y 

we shall have a new equation Y — 0. 

Since we know, from the relation x — —^ that the greatest 

2/ 



CHAP. XI. CONSEQUENCES OF PRINCIPLES. 359 

positive va'ue of y in the new equation corresponds to the least 
positive value of x in the given equation, it follows, that 

If we determine the sxq^erior limit of the positive roots of the 
equation Y = 0, its reciprocal will be the inferior limit of the 
positive roots of the given equation. 

Her.ce, if we designate the superior limit of the posiiive 
roots of the equation 1^= by L\ we shall have for the in- 
ferior limit of the positive roots of the given equation, — . 

Second^ If in the equation 

JT = 0, we make ar = — y, 
which gives the transformed equation Y^ = 0, it is clear that 
the positive roots of this new equation, taken with the sign 
— , will give the negative roots of the given equation; thercr 
fore, determining by known methods, the superior limit of the 
positive roots of the new equation Y' = 0, and designating this 
limit by Z''^, we shall have — L^^ for the superior limit, (nu- 
merically), of the negative roots of the given equation. 

Thirds If in the equation 

X = 0, we make x =. , 

y 

we shall have the derived equation Y^^ = 0. The greatest posi- 
tive value of y in this equation will correspond to the least 
negj^tive value (numerically) of x in the given equation. If, 
then, we find the superior limit of the positive roots of the 
equation Y^^ = 0, and designate it by Z^^^, we shall have the 

inferior limit of the negative roots (numerically) equal to — — ^ 

Co72 sequences deduced from the ijreceding Principles. 

First. 

287. Every eq^mtion in ivhich there are no variations in the signs^ 
that is, in which all the terms are positive^ must have all of its real 
roots negative; for, every positive number substituted for a:, will 
render the first member essentially positive. 



360 ELEMENTS OF ALGEBRA. [CHAP. XL 

Second. 

288 • Every complete equation^ having its terms alternately posi- 
tive and negative, must have its real rocs all positive ; for, every 
negative number substituted for x in the proposed equation, would 
render all the terms positive, if the equation be of an even de 
gree, and all of them negative, if it be of an odd degree. Hence, 
their sum could not be equal to zero iu either case. 

This principle is also true for every incomplete equation, in which 
there results, by substituting — y for x, an equation having all its 
terms affected with the same sign. 

Tliird. 

289. Every equation of an odd degree, the co-efficients of whicH 
are real, has at least one real root affected with a sign contrary to 
that of its last term. 

For, let 

^m _^ p^m-\ + . . . Txzh U=0, 

be the proposed equation ; and first consider the case in which 
the last term is negative. 

By making a: r= 0, the first member becomes — U. But by 
giving a value to x equal to the greatest co-efficient plus 1, or 
(A"+ 1), the first term o:^ ^vill become greater than the arith- 
metical sum of all the others (Art. 2S2), the result of this sub^ 
stitution will therefore be positive; hence, there is at least one 
real root comprehended between and ^+1, which root is posi- 
tive, and consequent!}' affected with a sign contrary to that of th© 
last term (277). 

Suppose now, that the last term is p>ositive. 

Making a: = in the first member, we obtain -f U for the result ; 
but by putting — (^4 1) i'l place of a:, we shall obtain a negu^ 
live result, since the first term becomes negative by this sab 
stitution ; hence, the equation has at least one real root com 
prehended between and — {K -\- 1), which is negative, oi 
affected with a sign contrary to tliat of the last XTm. 



CUAP. XI. I CONSEQUENCES OF PRINCIPLES. 361 

Fourth. 

290. Every equation of an even degree^ which involves only rea. 
CO ('/riL'ieiits, and of which the last term is negative, has at least two 
real roots, one jjosidve and the other negative. 

For, let — U be the last term ; making ar r= 0, there results 
— U. Now, substitute eilher X-\~ 1, or — (IC -{- 1), ^ being 
the greatest co-efficient in the equation. As m is an even number, 
the first term x^ will remain positive ; besides, by these substi- 
tutions, it becomes greater than the sum of all the others ; there- 
fore, the results obtained by these substitutions are both j^ositive^ 
or affected with a sign contrary to that given by the hypothesis 
X zzz \ hence, the equation has at least two real roots, one j^ositive, 
and comprehended between and A^4- 1, the other negative, and 
comprehended between and — {K -{- 1) (277). 

Fifth. 

291. 7/ an equation, involving only real co-efficients, contains imagi- 
nary roots, the number of such roots must he even. 

For, conceive that the first member has been divided by all the 
simple factors corresponding to the real roots; the co-efficients 
of the quotient will be real (Art. 246); and the quotient must also 
he of an even degree ; for, if it was mieven. by placing it equal 
to zero, we should obtain an equation that would contain at least 
one real root (289) ; hence, the imaginary roots must enter 
by pairs. 

11emat:k. — There is a property of the above polynomial quotient 
which belongs exclusively to equations containing only imaginary 
roots; viz., every such equation always remains positive for any 
real value substituted for x. 

For, by substituting for x, K -{-\^ the greatest co-efficient 
plus 1, we could always obtain a positive result; hence, if the 
polynomial could become negative, it would follow that when 
placed equal to zero, there ^ould be at least one real roo<i com- 



382 ELEilENT'S OF ALGEBRA. [CHAP. 2L 

prelieiided between ^"4- 1 and the number which would give a 
negative result (Art. 277). 

It also follows, that the last term of this polynomial must be 
positive, otherwise x = would give a negative result. 

Sixth. 

292i When the last term of an equation is 2^ositive, the 7imnber 
of its real positive roots is even ; and when it is negative^ tJue 
nuraher of such roots is uneven. 

For, first suppose that the last term is -f TI, or 2^ositive. Since 
by making x — 0, there will result + U, and by making x = K -\-l, 
the result will also be positive, it follows that and ^ + 1 
give two results affected with the same sign, and consequently 
(Art. 279), the number of real roots, if any, comprehended be- 
tween them, is even. 

When the last term is — U, then and -^4- 1 give two 
results aifected with contrary signs, and consequently, they com- 
prehend either a single root, or an odd number of them. 

Tlie converse of this proposition is evidently true. 

Descartes' Rule. 

293, An equation of any degree tvhatever, cannot have a greater 
number of j^ositive roots than there are variations in the signs of 
its terms, nor a greater number of negative roots than there are 
permanences of these signs. 

A variation is a change of sign in passing along the terms. A 

permanence is when two consecutive terms have the same sign. 

In the equation 

a; — a = 0, 

there is one variation, and one positive root, a: = a. 

And in the equation x -\-b z=0, there is one permanence, and 
one negative root, x = — b. 

If these equations be multiplied together, member by member, 
there will resul!; an equation of the second degree, 
a;2 — a 



+ b 



"' I = 0. 



I 



CHAP. XI.j DESCARTES' RULE, 863 

If a is less .han h^ the equation will be of the first form 
(Art. 117); and if a ^ b, the equation will be of the second 
form ; that is, 

a < 6 gives x"^ -f- 2jt?a; — g = 0, 
and a > 6 " x^ — 2px — q = 0. 

In the first case, there is one permanence and one variation, 
and in the second, one variation and one permanence. Since 
in either form, one root is positive and one negative, it fol- 
lows that there are as many positive roots as there are 
variations, and as many negative roots as there are perma- 
nences. 

The proposition will evidently be demonstrated in a general 
manner, if it be shown that the multiplication of the first mem- 
ber of any equation by a factor x — a, corresponding to a posi- 
tive root, introduces at least one variation^ and that the multi- 
plication by a factor x -f a, corresponding to a negative root, 
introduces at least one permanence. 

Take the equation, 

.n which the signs succeed each other in any manner whatever. 
By multiplying by x — a, we have 

■2 ± ... ±U 
^Ta 

The co-efficients which form the first horizontal line of this 
product, are those of the given equation, taken wHh the same 
signs ; and the co-efficients of the second line are formed from 
those of the first, by multiplying by a, changing the signs, and 
advancing each one place to the right. 

■Vow, so long as each co-efficient in the upper line is greater 
than the corresponding one in the lower, it will determine the 
sign of the total co-efficient; hence, in this case there wiil be, 
from the first term to that preceding the last, inclusively, the 
same variations and the same permanences as in the proposed 
equation ; but the last term zp Ua having a sign contrary to that 
which immediately precedes it, there must De one more varia- 
tion ihan in the proposed equation. 



— a 



■B 

■ Aa 



zpBa 



.1 



u.-'- 



BQi ELEMENTS OF ALGEBRA. [CHAP. XI. 

WHen a co-efficient in the lower line ^'s affected with a sign 
contrary to the one corresponding to it in the upper, and id 
also greater than this last, there is a change from a perma 
nence of sign to a variation ; for the sign of the term in whicn 
this happens, being the same as that of the inferior co-efficient, 
must be contrary to that of the preceding term, which has 
been supposed to be the same as that of the superior co-effi- 
cient. Hence, each time we descend from the upper to the 
lower line, in order to determine the sign, there is a variation 
which is not found in the proposed equation ; and if, after 
passing into the lower line, we continue in it throughout, we 
shall find for the remaining terms the same variations and the 
same permanences as in the given equation, since the co-efficients 
of this line are all affected with signs contrary to those of the 
primitive co-efficients. This supposition would therefore give us 
one variation for each positive root. But if we ascend from 
the lower to the upper line, there may be either a variation 
or a permaneiice. But even by supposing that this passage pro- 
duces permanences in all cases, since the last term ^ Ua fc/ms 
a part of the lower line, it will be necessary to go once -.nore 
from the upper* line to the lower, than from the lower to the 
upper. Hence, the new equation must have at least one more 
variation than the 2y^'oposed ; and it will be the same for each 
positive root introduced into it. 

It may be demonstrated, in an analogous manner, that the 
multiplication of the first member hy a factor x ■\- a^ correspond- 
ing to a negative root, would introduce one permanence more. 
Hence, in any equation, the number of positive roots cannot be 
greater than the number of variations of signs, nor the number 
ef negative roots gi eater than the number of permanences. 

Consequence. 

294. When the roots of an equation are all real, the number 
of positive roots is equal to the nvmber of variations^ and the nutr^ 
her of negative roots to the number of permanences. 



CHAP. XT.J DESCARTES' RULE. 865 

For, let m denote the degree of the equation,, n the number 
of variations of the bigns, p the number of permanences ; then, 

m = n -\- p. 
IMoreover, let n' denote the number of positive roots, and p' 
Uie number of negative roots, we shall have 

m = n' -\- p^ \ 
whence, n -{-p = n' -\- p% or, n — n' z=p^ —p. 

Now, we have just seen that n/ cannot be > n, nor can it be 
less, since p^ cannot be > ^ ; therefore, we must have 
iV = n, and p^ z=: p. 
Remark. — When an equation wants some of its terms, we can 
often discover the presence of imaginary roots, by means of the 
above rule. 

For example, take the equation 

X'^ -\- pX -\- q :=:z()^ 

p and q being essentially positive ; introducing the term which 
is wanting, by affecting it with the co-efficient ± ; it becomes 

X^ ± . X'^ -\- pX -\- q =: 0. 

By considering only the superior sign, we should obtain only 
permanences, whereas the inferior sign gives two variations. This 
proves that the equation has some imaginary roots ; for, if they 
were all three real, it would be necessary, by virtue of the supe- 
rior sign, that they should be all negative, and, by ■ virtue of the 
inferior sign, that two of them should be positive and one nega- 
tive, which are contradictory results. 

We can conclude nothing from an equation of the form 
x"^ — p)x -\- q - ; 
for, introducing the term dt . a:^, it becomes 

x"^ dsz . x"^ -- px -\- q =z 0, 
which contains one permanence and two variations, whether we 
take the superior or inferior sign. Therefore, this equation may 
have its three roots real, viz., two positive and one negative ; 
or, two of its roots may be imaginary and one negative, since 
its last term is positive (Art. 292). 



366 ELEMENTS OF ALGEBRA. LCHAP. XL 

Of the coinmensurahle Roots of Numerical Eq^iiatioi.s. 

295. Every equation in which the co-efficients aie whole num- 
bers, that of the first term being 1, will have whole numbers 
only for its commensurable roots. 

For, let there be the equation 

x^ + Px"^-^ + Qx^-"^ + . . . -[- Tx-\-U=^0', 
in which P, Q , . . T, U, are whole numbers, and suppose that 

It were possible for one root to be an irreducible fraction — . 

Substituting this fraction for x, the equation becomes 

whence, multiplying both members by b'^-^, and transposing, 

%- = — Fa"^'^ — Qa'^^b — . . . _ Tah"^-^ — Ub"^-^. 
b 

But the second member of this equation is composed of 
the sum of entire numbers, while the first is essentially frac- 
tional, for a and b being prime with respect to each other, a*" 
and b will also be prime wdth respect to each other (Art. 95), 
and hence this equality cannot exist; for, an irreducible frac- 
tion cannot be equal to a whole number. Therefore, it is im- 
possible for any irreducible fraction to satisfy the equation. 

Now, it has been shown (Art. 262), that an equation con- 
taining rational, but fractional co-efficients, can be transformed 
into another in w^hich the co-efficients are whole numbers, 
that of the first term being 1. Hence, tJie search for commensu- 
rable roots, either entire or fractional, can always be reduced to 
that for entire rootfi. * 

296. This being the case, take the general equation 

and let a denote, any entire number, positive or negative, which 
will satisfy it. 

Since a is a oot, we shall have the equation 
tt« 4 Pa"^^ V ... + i2a3 -f- ,S'a2 -^ Ta-\- U= . (1). 



CHAP. XI.] COMMENSURABLE ROOTS OF EQUATIONS. 8G7 

Now replace a by all the entire numbers, positive and negative, 
between 1 and the limit +Z, and between — 1 and — U' '. those 
which verify the above equality will be roots of the equation. 
But these trials being long and troublesome, we will deduce from 
equation (1), other conditions equivalent to this, and more easily 
applied. 

Transposing in equation (1) all the terms except the last, and 
dividing by a, we have, 
U 



— — a" 



a 



PaJ^'' - . . . ~Ra? - Sa- T • - - (2). 



Now, the second member of this equation is an entire number ; 

hence, — must be an entire number : therefore, the entire roots of 

the equation are comprised among the divisors of the last term. 
Transposing — T in equation (2), dividing by a, and making 

— -h r = r, we have, 
a 

T 

— = — ^"^2 _ p^m_3 _ ^ _jia — S - - - . (3). 

a ^ ' 

T' 

The second member of this equation being entire, — , that is, 

a 

the quotient of 

■^+^ by a, 

IS an entire number. 

Transposing the term — >S and dividing by a, we have, by 
supposing 

T 
a 

— = - a^-^ - Pa*"-* - . . . - i2 - - - (4). 

a ^ ' 

Tlie second member of this eqiiati m jeing entire. — , that is, 

tht quotient of 

T 

— + 5 by «, 

%i an entire number. 



368 ELEMENTS OF ALGEBRA. [CHAP, XI. 

By continuing to transpose the terms of the second member 
into the first, we shall, after m — 1 transformations, obtain an 
equation of the form, 

a 

Then, transposing the term — P, dividing by a, and making 

O' P' P^ 

— + P = P\ we have —:==-!, or h 1 = 0. 

a a a 

Tliis equation, which results from the continued transforma^ 
tions of equation (1), expresses the last condition which it is 
requidte for the entire number a to fulfd, in order that it may 
be known to be a root of the equation. 

297. From the preceding conditions we conclude that, when 
an entire number a, positive or negative, is a root of the given 
equation, the quotient of the last term, divided hy a, is an 
entire number. 

Adding to this quotient the co-efRcient of rr^, the sum will 
he exactly divisible by a. 

Adding the co-efficient of x"^ to this last quotient, and again 
dividing by a, the new quotient must also be entire; and so on. 

Finally, adding the co-efficient of the second term, that is, of 
a,"*""^, to the preceding quotient, the quotient of this sum divided 
by a, must be equal to — 1 ; hence, the result of the addition of 
1, which is the co-efficient of x™, to the preceding quotient, must 
he equal to 0. 

Every number which will satisfy these conditions will be a 
root, and those which do not satisfy them should be rejected. 

All the entire roots may be determined at the same time, 
by the following 

RULE. 

After having determined all the divisors of the last term, ivrite 
those which are comprehended between the limits 4* L and — \J^ 
upon the same horizontal line ; then underneath these divisors write 
the quotients of the last term by each of them. 



CHAP. XI.] COMMENSUKABLE ROOTS. 369 

Add the co-efficient of x^ to each of these quotients^ and write 
the S2ims underneath the quotients w/tich correspond to them. 
Then divide these sums by each of the divisors^ and write the quo- 
tients underneath the corresponding sums, taking care to reject the 
fractional quotients and the divisors which produce them ; and 
so on. 

When there are terms wantmg in the proposed equation, 
their co-efficients, Avhich are to be regarded as equal to 0, must 
be taken into consideration. 

EXAMPLES. 

1. What are the entire roots of the equation, 

A superior limit of the positive roots of this equation (Art.. 
284), is 13 + 1 =: 14. The co-efficient 48 need not be con- 
sidered, since the last two terms can be put under the form 
IQ {x — 3) ; hence, when a; > 3, this part is essentially positive. 

A superior limit of the negative roots (Art. 286), is 

-(1+/48), or -8. 

Therefore^ the divisors of the last term which may be roots,, 
are 1., 2, 3, 4, 6, 8, 12 ; moreover, neither + 1, nor — 1, will 
satisfy the equation, because the co-efficient — 48 is itself greater 
than the sum of all the others : we should therefore try only 
the positive divisors from 2 to 12, and the negative divisors from 
— 2 to — 6 inclusively. 

By observing the rule given above, we have 
12, 8, 6, 4, 3, 2, - 2, - 3, - 4, - 6 



- 4, - 6, - 8, - 12, - 16 
-f 12, 4 10, + 8, + 4, 0^ 

f 1, .., .., + 1, 

-12, .., .., -12, -^3 

- 1, .., .., — 3, 

- 2, .., .., - 4, 

••» ••! "J 1> 



- 24, -f 24, + 16, + 12, + 8 

- 8, + 40, -f 32,- + 28, + 24 

- 4, -20, 
-17, -33, 



24 



., - ^, 


- 4 


., - 20, 


- 17 


.., + 5, 




„+ 4, 


.. 


., - 1, 


„ 



870 ELEMENTS OF ALGEBRA. LCHAP. XI. 

The first line .contains the divisors, the second contains the 
quotients arising from the division of the last term — 48, by 
each of the divisors. The third line contains these quotients, each 
augmented by the co-efficient + 16 ; and the fourth, the quotienta 
of these sums by each of the divisors ; this second condition 
excludes the divisors +8, +6? ^^^ — 3« 

The fifth contains the preceding line of quotients, each aug 
mented by the co-efficient — 13, and the sixth contains the quo 
tients of these sums by each of the divisors ; the third condition 
excludes the divisors 3, 2, — 2, and — 6. 

Finally, the seventh is the third line of quotients, each aug 
mented by the co-efficient — 1, and the eighth contains the quo- 
tients of these sums by each of the divisors. The divisors -f- 4 
and — 4 are the only ones which give — 1 ; hence, -f 4 and 
— 4 are the only entire roots of the equation. 

In fact, if we divide 

X* — x^ — 13.t2 + 16a; — 48, 

by the product (a; — 4) (a; + 4), or x^ — 16, the quotient wi.] 
be a:2 — ir -f 3, which placed equal to zero, gives 



1 1 y TT 



therefore, the four roots are 



4, -4, 1 + l/3Tr and | - |/ 



2 ■ 2V 2 

2. What are the entire roots of the equation 

x^-bx^-j- 25a; -21 =0? 
8. What are the entire roots of the equation 

15a;s - 19a;* + 6x^ -f 15a;2 - 19a; + 6 = ? 

4. What are the entire roots of the equaton 

9a;6 f 30a:5 _{_ 22a;* -|- lOa;^ + 17a'2 - 20a; -f- 4 =r ? 



11. 



CHAP. xiJ stuem's theorem. 871 

Sturm^s Theorem, 

298. The object of this theorem is to explain a method of de. 
termining the number and places of the real roots of equations 
involving but one unknown quantity. 

Let X=:zO ... - (1), 

represent an equation containing the single unknown quantity x ; 
X being a polynomial of the m*^ degree with respect to a;, the 
co-efficients of which are all real. If this equation should have 
equal roots, they may be found and divided out as in Art. 269, 
and the reasoning be applied to the equation which would result. 
We will therefore suppose X = to have no equal roots. 

299. Let us denote the first derived polynomial of X by Xj, 
and then apply to X and X^ a process similar to that for find- 
ing their greatest common divisor, differing only in this respect, 
that instead of using the successive remainders as at first ob- 
tained, we change their signs, and take care also, in preparing for 
the division^ neither to introduce nor reject any factor except a 
positive one. 

If we denote the several remainders, in order, afler their signs 
have been changed, by X^, JTg . . . X,, which are read X second, 
X third, drc, and denote the corresponding quotients by Q^t Qt 
. . Qt^iy we may then form the equations 

X=X,^,~X, .... (2). 



X,_, = X.§. - X.+, J. ■ - - (3). 
I 

Since by hypothesis, X = has no equal roots, no common 
divisor can exist between X and X, (Art. 2G7). The last r^ 
mainder — A*",, will therefore he different from zero, and indt- 
pendent of x. 



372 ELEMENTS OF ALGEBRA. [CHAP. XI. 

300. Now, let us suppose that a number p has been substi 
tuted for x in each of the expressions X, Xj, Xj . . . X^i ; 
and that the signs of the results, together with the sign of X,, 
are arranged in a line one after the other : also that another 
number q, greater than p, has been substituted for x, and the 
rigns of the results arranged in like manner. 

Then will the number of variations in the signs of the first 
arrangemejit, diminished by the number of variations in those of 
the second, denote the exact number of real roots comprised be- 
tween p and q. 

301. The demonstration of this truth mainly depends upon 
the three following properties of the expressions X, Xj . . X„, &c. 

I. If any number be: substituted for x in these expressions, it is 
impossible that any two consecutive ones can become zero at the 
same time. 

For, let X^i, X„, X^+i, be any three consecutive expressions. 
Hien among equations (3), we shall find 

from which it appears that, if X^i and X„ should both become 
for a value of a:, X„+i would be for the same value ; and 
since the equation which follows (4) must be 

we shall have X„+2 = for the same value, and so on until 
we should find X, = 0, which cannot be ; hence, X^i and X, 
cannot both become for the same value of x. 

II. By an examination of equation (4), we see that if X„ be- 
comes for a value of a;, X^. and X,+i must have contrary 
signs; that is, 

If any one of the expressions is reduced to by the substi- 
tution of a value for x, the preceding and following' ones will 
have contrary signs for the same value. 



i 



CHAP, xi.j Sturm's theorem. 373 

III. Let us substitute a -f « for x in tho expressions X and 
Xi, and designate by U and Ux what tney respectively become 
under this supposition. Then (Art. 264), we have 



u 



U =A -\-A'u + A'' —■ 4- &c. 



C7"i = ^1 + A\u + A'\— + &c. 

in which A^ A\ A'\ &c., are the results obtained by the sub- 
stitution of a- for rr, in X and its derived polynomials ; and 
^1, A\^ &c., are similar results derived from X^, If, now, a be 
a root of the proposed equation X = 0, then ^ =z 0, aid since 
A' and A-^ are each derived from Xj, by the substitution of 
a for a:, w^e have A^ = A^, and equations (5) become 

U=A'u + A"^^ + ^. . . . (6). 



CTi = ^' + A\u + &c. 

Now, the arbitrary quantity u may be taken so small that 
the signs of the values of U and Ui will depend upon the 
signs of their first terms (Art. 276) ; that is, they will be alike 
when u is positive, or when a -{- u is substituted for x, and un^ 
like when u is negative or when a — u is substituted for x. 
Hence, 

If a number insensibly less than one of the real roots of 

X. =^ be substituted for x in X and Xj, the results will have 

contrarij signs; and if a number insensibly greater than this root 
be substituted^ the results will have the same sign. 

302i Now, let any number as k, algebraically less, that is, 
nearer equal to — oo, than any of the real roots of the seveial 
equations 

X=0, X, = . . . X,_, = 0, 

be substituted for x in the expressions X, Xj, Xj, &;c., and the 
signs of the several results arranged in order ; then, let x be 
increased by insensible degrees, until it becomes equal to h^ 
the least of all the roots of the equations. As there is na 



374 ELEMENTS OF ALGEBRA. [CHAP. XI. 

root of either of the equations between h and ^, none of the 
signs can change while x is less than h (Art. 277), and the 
number of variations and permanences in the several sets of 
results, will remain the same as in those obtained by the first 
substitution. 

When X becomes equal to A, one or more of the expressions 
X, JT, &c., will reduce to 0. Suppose X„ becomes 0. Then, 
as by the first and second properties above explained, neither 
X,»_i nor X„+i can become at the same time, but must have 
contrary signs, it follows that in passing from one to the other 
(omitting X„ =: 0), there will be ojie and only one variation ; 
and since their signs have not changed, one must be the same 
as, and the other contrary to, that of Jr„, both before and after 
it becomes ; hence, in passing over the three, either just before 
X„ becomes or just after, there is one and only one variation. 
Therefore, the reduction of X^ to neither increases nor di- 
minishes the number of variations ; and this will evidently be 
the case, although several of the expressions Xj, Xj, &c., should 
become at the same time. 

If X =1 h should reduce X to 0, then h is the least real root 
of the proposed equation, which root we denote by a; and 
since by the third property, just before x becomes equal to a, 
the signs of X and Xj are contrary, giving a variation, and just 
after passing it (before x becomes equal to a root of X^ =: 0), 
the signs are the same, giving a permanence instead, it follows 
that in passing this root a variation is lost. 

In the same way, increasing x by insensible degrees from 
X z=a -^ u until we reach the root of X = next m order, it 
is plain that no variation will be lost or gained in passing any 
of the roots of the other equations, but that in passing this 
root, for the same reason as before, another vaiiation will be 
lost, and so on for each real root between Jc and the number 
last substituted, as g, a variation will be lost until a; has been 
increased beyond the greatest real root, when no more can be 
lost or gained. Hence, the excess of the number of variations 



CEAP. XT.] Sturm's theorem. 375 

obtained by the substitution of k over those obtained by the 
substitution of g^ will be equal to the number of real roots 
comprised between k and g. 

It is evident that the same course of reasoning will apply 
when we commence with any number p^ whether less than all 
the loots or not, and gradually increase x until it equals any 
other number q. The fact enunciated in Art. 299 is therefore 
established. 

303. In seeking the number of roots comprised between p and g, 
bhould either p or q reduce any of the expressions Xj, JTj, &c., 
to 0, the result will not be affected by their omission, since 
the number of variations wiU be the same. 

Should p reduce X to 0, then /> is a root, but not one of those 
sought ; and as the substitution o^ p -{- u will give X and X^ 
the same sign, the number of variations to be counted will not 
be affected by the omission of X = 0. 

Should q reduce X to 0, then q is also a root, but not one 
of those sought ; and as the substitution o^ q — u will give X 
and X, contrary signs, one variation must be counted in passing 
from X to Xj. 

304. If in the application of the preceding principles, we ob- 
serve that any one of the expressions Xj, X^ . . . &c., X„ for 
instance, will preserve the same sign for all values of x in 
passing from p to q^ inclusively, it will be unnecessary to use 
the succeeding expressions, or even to deduce them. Tor, as 
X„ preserves the same sign during the successive substitutions, 
it is plain that the same number of variations will be lost 
among the expressions X, Xj, &;c, . , . ending with X„ as among 
all including X,. Whenever then, in the course of the division, 
it is found that by placing any of the remainders equal to 0, 
an equation is obtained with im£:.giuary roots only (Art. 291), 
it will be useless to obtain any of the succeeding remainders. 
This principle will be found very useful in the solution of 
numerical CAamples. 



376 ELEMENTS OF ALGEBRA. [CHAP. XI 

305. As all the real roots of the proposed equation are neces- 
sarily included between — oo and + oo, we may, by ascertain- 
ing the number of variations lost by the substitution of these, 
in succession, in the expressions X, Xj . . . X„, . . &c., readily 
determine the total number of such roots. It should be ob- 
served, that it will be only necessary to make these substitu- 
tions in the first terms of each of the expressions, as in this 
case the sign of the term will determme that of the entire ex- 
pression (Art. 282). 

Having found the number of real roots, if we subtract this 
number from the highest exponent of the unknown quantity, the 
remainder will be the number of imaginary roots (Art. 248). 

306. Having thus obtained the total number of real roots, 
we may ascertain their places by substituting for x^ in succes- 
sion, the values 0, 1, 2, 3, &c., until we find an entire num- 
ber which gives the same number of variations as -f- oo. This 
will be the smallest superior limit of the positive roots in entire 
numbers. 

Then substitute — 1, — 2, &;c., until a negative number is 
obtained which gives the same number of variations as — oo. 
This will be, numerically, the least superior limit of the 
negative roots in entire numbers. Now, by commencing with 
this limit and observing the number of variations lost in passing 
from each number to the next in order, we shall discover how 
many roots are included between each two of the consecutive 
numbers used, and thus, of course, know the entire part of eacK 
root. Tlie decimal part may then be sought by some of thf 
known methods of approximation. 

EXAMPLES. 

1. Let &x'3 — 6^— 1 = =X. 

The first derived poiynomiial (Art. 264), i* 
24a;2 - 6, 



OHAP. XI.] Sturm's theorem. 377 

and since we may omit the positive factor 6^ (without affecting 
the sign, we may write 

Dividing X by Xi, we obtain for the first remainder, —4a: — 1. 
Changing its sign, we have 

4.x-\-\ =X^. 

Multiplying X^ by the positive number 4, and then dividing 
by X2, we obtain the second remainder — 3 ; and by changing 
its sign 

+ 3 rz X3. 

The expressions to be be used are then 

X = 8^3 - 6a; - 1, X, = 4x^ - 1, X, = 4:X-\-l, X,= -{- 3. 

Substituting — oo and then -h «>, we obtain the two following 
arrangements of signs : 

— + — 4- 3 variations, 

+ + + + 

there are then tlirse real roots. 

If, now, in the same expressions we substitute and + 1, 
and then and — 1, for a:, we shall obtain the three following 
arrangements : 

For X = ■\- \ -\ — \- -\ — I- variations, 

" x= h+ 1 " 

" x= -I 1 h 3 " 

As X z=: -\- \ gives the same number of variations as -^ 00, 
and X z=z — \ gives the same as — od, + 1 and — 1 are the 
smallest limits in entire numbers. In passing from — 1 to 0, 
two variations are lost, and in passing from to + 1, on« 
variation is lost ; hence, there are two negative roots bet'-veeu 
— 1 and 0, and one positive root between and -\- \. 

2. Let 2a;* - IZx"^ -]- lOo: - 19 := 0. 



S78 ELEMENTS OF ALGEBRA. [CHAP. XI 

If we deduce X, Xj, and X, we have the three expressions 
X = 2x^- ISx^ + 10a; - 19, 
Xi = 4a;3 - 13;r + 5, 
X, == 13^-2 - 15^ + 38. 

If we place X^ r= 0, we shall find that both of the roots of 
the resulting equation are imaginary ; hence, X^ will be positive 
for all values of x (Art. 290). It is then useless to seek for 
Xg and X4. 

By the substitution of — gc and + oo in X, Xi, and Xj, we 
obtain for the first, tivo variations, and for the second, none; 
hence, there are two real and two imaginary roots in the 
proposed equation. 

3. Let a;3 — 5^2 _|_ 8^ _ 1 _ q. 

4. a;* — a;3 — 3^2 + ;i-2 _ ^ _ 3 _ q. 

5. a;5 - 2^3 + 1 = 0. 
Discuss each of the above equations. 

307» In the preceding discussions we have supposed the 
equations to be given, and from the relations existing between 
the co-efficients of the different powers of the unknown quan- 
tity, have determined the number and places of the real roots; 
and, consequently, the number of imaginary roots. 

In the equation of the second degree, we pointed out the 
relations which exist between the co-efficients of the different 
powers of the unknown quantity when the roots are real, 
and when they are imaginary (Art. 116). 

Let us see if we can indicate corresponding relations among 
the co-efficients of an equation of the third degree. 

Let us take the equation, 

a;3 + Pa-2+ Qx+ ?7=0, 

and by causing the second term to disappear (Art. 263), it 
will Lake the form, 

x^ -\- px -\- q — 0^ 



CHAP, xi/ cardan's rule. 379 

Hence- we have 

X = x^ -{-px-^-q, 

X, = — 2px — Zq^ 
X^= - 4^3 - 21 q\ 

It order that all the roots be real, the substitution of oo for 
X in the above expressions must give three permanences; and 
the substitution of — oo for x must give three variations. But 
the first supposition can only give three permanences when 
— 42j3 - 27g2 > ; 

hat is, a positive quantity, a condition which requires that p be 
Qegative. 

If, then, p be negative, we have, for a: =r oo, 

— 4p3 — 21 q^ > ; that is, positive : 

or, 4p"^ + 27^2 <^ ; that is, negative : 

hence, T ~^ o^ ^ ^' ^^^^^ requires that p be 

negative, and that ^ > -r- ? conditions which indicate that the 
roots are all real. 

Cardaii's Rule for Solving Cubic Equations. 

308. First, free the equation of its second term, and we 
have the form, 

x^ -^px -{-q = (1). 

Take x = y -\- z\ 

then x'^ = y^ + 2;^ 4- 3?/0 (y + ^) ; 

or, ])y transposing, and substituting x for y + 2, we have 
^.3 _ St/z . .i - (y3 + 23) = - - - (2) ; 
and by comparing this with equation (1), we have 
— Syz — p ] and y3 4 ^3 _ — q^ 



380 . ELEMENTS OF ALGEBRA. 

From the 1st. we have 

3— ^^ 

which, being substituted in the second, gives 

^3 



[CHAP. XL 



r 



21f 



= -?; 



cr clearing of fractions, and reducing 



2/6 -I- qy2 



27' 



Solving this trinomial equation (Art. 124), we have 
and the corresponding value of z is 



V ~ 2 V 4 + 27- 



But since x =z y -{- z, we have , 

"V[-*+vf^)]+y[-f-v/(?^) 

This is called Cardaii's formula. 

By examining the above formula, it will be seen, tha.. it ia 
inapplicable to the case, when the quantity 

4 "^27' 

under the radical of the second degree, is negative ; and hence 
is applicable only to the case where two of the roots are imag 
inary (Art. 307). 

Having found the real root, divide both members o^ the giveu 
equation by the unknown quantity, minus this root (Art. 247) ; 
the result will be an equation of the second degree, the roots 
of which may be readily found. 



CHAP. xi.I horner's method. ZSl 



EXAMPLES. 



1. What are the roots of the equation 



Ans. 4, l+y^=^ l-V'-^' 

2. What are the roots of the equation 

a;3 _ 9^2 _|. 28^ = 30 1 

Ans. 3, 3+y^^^, 3-y-l. 

3. What are the roots of the equation 

a;3 _ 7^2 4. 14^ ^ 20 ? 

^ws. 5, i4-y^=^, 1 -y-T. 

Preliminaries to Horner^ s Method. 

309. Before applying the method of Horner to the solution 
of numerical equations, it will be necessary to explain, 

1st. A modification of the method of multiplication, called 
the method by Detached Co-efficients : 

2d. A modification of the method of division, called, also, the 
method by Detached Co-efficients : 

3d. A second modification of the method of division, called 
Synthetical Division : and, 

4th. The application of these methods of Division in the 
Transformation of Equations, 

Multiplication hy Detached Co-efficients, 

310. When the multiplicand and multiplier are both homo- 
geneous (Art. 26), and contain but two letters, if each be ar- 
ranged according to the same letter, the literal part, in the 
several terms of the product, may be written immediately, since 
the exponent of the leading letter will go on decreasing from 
left to right by a constant number, and the sum of the exponents 
of both letters will be the same, in each of the terms. 



382 ELEMENTS OF ALGEBRA, [CHAP. XI. 



EXAMPLES. 



1, Let it be required to multiply 

ar3 _|_ x^y + xy"^ -\- y"^ by ar — y. 
Since x^ x x = x^, the terms of the product will be of th« 
4th degree, and since the exponents of x decrease by 1, and 
those of y increase by 1, we may write the literal parts thus, 
a:*, x^y, x^y% xy^, y*. 
In regard to the co-efficients, we have, 
Co-efficients of multiplicand, - - - 1 + 1-f-l-rl. 
" " multiplier, - 1—1 

14-1 + 1 + 1 



co-efficients of the product, - - 1+0 + + 0—1; 

aixi writing these co-efficients before the literal parts to which 
they belong, we have 

a:* + . a:3y + . x^y^ + . xy^ — y^ = x* — y\ 

2. Multiply 2a3 — Sab^ + 563 ^y 2a^ - 5b\ 

In this example, the term a'^b in the multiplicand, and ab in 
the multiplier, are both wanting ; that is, their co-efficients are 
0. Supplying these co-efficients, and we have, 

Co-efficients of multiplicand, - 2 + — 3 + 5 
" " multiplier, - - 2 + — 5 

4 + 0- 6 + 10 

-10- + 15-25 



co-efficients of the product, - - 4 + — 16+10 + 15—25. 
Henoe, the product is, 4a5 - l^a^^ + lOa^^ + 15a5* - 25*». 

3. Multiply x^ — Sx^ + Sx — 1 by x^—2x-^l. 

4. Multiply 2/2 _ ^Qj _l_ Qj2 -^y y2 _|. ^^ ,^ ^^2^ 

Remark. — The method by detached co-efficients is also appli- 
cable to the case, in which the multiplicand and multiplier cotv- 
tain but a single letter. The terms whose co-efficients are zero 
must be supplied, when wanting, as in the previous examples. 



CHAP. XI.' DETACHED CO EFFICIENTS. 883 

EXAMPLES. 

1. What is the product of a* + 3a2 + I by a^ — 3 1 

2. What is the product of b^ ~ 1 by 6 -h 2 ? 

Division by Detached Co-efficients, 

311. When the dividend and divisor are both homogeneoii$ 
and contain but two letters, the division may be performed by 
means of detached co-efficients, in' the following manner : 

1. Arrange the terms of the dividend and divisor according 
to a common letter. 

2. Subtract the highest exponent of the leading letter of the divi- 
sor from the exponent of the leading letter of the dividend, and 
the remainder will be the exponent of the leading letter of 
the quotient. 

3. The exponents of the letters in the other terms follow 
the same law of increase or decrease as the exponents in the 
corresponding terms of the dividend. 

4. Write down for division the co-efficients of the different 
terms of the dividend and divisor, with their respective signs, 
supplying the deficiency of the absent terms with zeros. 

5. Then divide the co-efficients of the dividend by those of 
the divisor, after the mariner of algebraic division, and prefix the 
several quotients to their corresponding literal parts. 

EXAMPLES. 



2. Divide 


8a5 - 4:a'x - 


- 2a3a;2 -h 22a;3 by 


4a2 - x\ 


The 


literal 


part will be 








< 


a^x^ ax^^ x^ ; 




and fcr the 


numerical co- 


■efficients, 








8-4- 


-2+1 4+0- 


u 






8 + 0- 


-2 2-1 








-4 


+ 1 








-4 


+ 1 

















384 ELEMENTS OF ALGEBRA. [CHAP. XI. 

hence, the true quotient is 2a^ — a^x] the co-efficients after — 1, 
being each equal to zero. 

3. Divide x^ — Sax^ — Sa^x^ + ISa^x — 8a* hj x^ — 2ax —2a*, 

4. Divide 10a* — 27a^x + Ma'^x'^ — ISax^ — 8x^ by 2a? 
— Sax -f 42-2. 

Remark. — The method by detached co-efficients is also appli- 
cable to all cases in which the dividend and divisor contain but 
a single letter. The terms whose co-efficients are zero, must be 
supplied, when wanting, as in the previous examples. 

EXAMPLES. 

1. Let it be required to divide 

Ga"^ — 96 by 3a — 6. 
The dividend, in this example, may be written under the form, 

6a* -f . a3 + . a2 + . a — 96a0. 
Dividing a* by a, we have a^ for the literal part of the 
first term of the quotient ; hence, the form of the quotient is 
a^, a^, a, a°. 
For the CO- efficients, we have, 

6-f0 + + 0-96||3-6 

6 — 12 2 + 4 + 8 + 16 quotient ; 

hence, the true quotient is, 

2a3 + 4a2 + 8a + 16. 

it 

Synthetical Division. 

312. In the common method of division, each term of the 
divisor is multiplied by the first term of the quotient, and the 
products subtracted from the di\ddend; but the subtractions are 
performed by first changing the sign of each product, and then 
adding. If, therefore, the signs of the divisor were first changed, 
we should obtain the same result by adding the products, instead 
of subtracting as before, and the same for any subsequent oper- 
ation. 



CHAP XI.] SYNTHETICAL DIVISION. 385 

By this process, the second dividend would be the same aa 
by the common method. But since the second term of the quo 
tient is found by dividing the first term of the second dividend 
by the first term of the divisor ; and since the sign of the latter 
has been changed, it follows, that the sign of the second term 
of the quotient will also be changed. 

To avoid this change of sign, the sign of the first term of the 
divisor is left unchanged, and the products of all the terms of 
the quotient by the first term of the divi"=:or, are omitted ; be 
cause, in the usual method, the first term.i in each successive 
dividend are cancelled by these products. 

Having made the first term of the divisor 1 before commene 
lug the operation, and omitting these several products, the co-effi- 
cient of the first term of any dividend will be fae co-efficient of the 
succeeding term of the quotient. Hence, the co-efficients in the 
quotient are, respectively, the co-efficients of the first terms of 
the successive dividends. 

The operation, thus simplified, may be f-irther abridged by- 
omitting the successive additions, except so much only as may 
be necessary to show the first term of each dividend ; and also, 
by writing the products of the several terms of the quotient by 
the modified divisor, diagonally, instead of horizontally, the first 
product falling under the second term of the dividend. 

Hence, the following 

RULE. 

I. Divide the divisor and dividend hy the co-efficient of the first 
term of the divisor^ when that co-efficient is not 1. 

II. Write, in a horizontal line, the co-efficients of the dividend^ 
with their proper signs, and place the co- efficients of the divisor y 
with all their signs changed, except the first, on the right. 

III. Divide as in the method hy detached co-efficients, except that 
iin term of the quotient is multijylied by the fi.rst term of the divi- 
sor, and that all the products are written diagonally to the right, 
under the terms of the dividend to which they correspond. 

25 



386 ELEMENTS OF ALGEBRA. [CHAP. XI. 

IV. The first term of the quotient is the same as that of the 
dividend ; the second term is the sum of the numbers in the second 
column ; the third term^ the sum of the numbers in third column, 
mid so on, to the right. 

V. When the division can be exactly made, columns will he found 
at the right^ whose sums will be zero : when the division is not 
exact, continue the operation until a suffi,cient degree of approxi- 
mation is attained. Having found the co-efficients, annex to them 
the literal parts. 

EXAMPLES. 

1. Divide 

a^ — 5a*a: + lOa^x"^ — lOa^x"^ -|- 5aa;* — x^ by a^ — 2a:F -f ^. 

1 - 5 + 10 - 10 + 5 - 1 II 1 + 2 - 1 

2- 6+ 6-2 i_3^.3^i 

- 1 + 3-3 + 1 



1 _ 3 + 3 _ 1 0. 

Hence, the quotient is 

a3 — Za'^x 4- 3aa:2 __ ^\ 

Remark. — The first term of the divisor being always 1, need 
not be written. The first term of the quotient is the same as 
that of the dividend. 

2. Divide 
««-5a;5+15a;*-242;34-27:c2_i3a._{_5 \^j a;*-2a;34-4rc2-2ar-f 1. 

l_54_15_24 + 27-13 + 5|Il + 2-4 + 2-l 

+ 2-6 + 10 '' rry+5 

- 4 + 12-20 

+ 2 - 6 + 10 

- 1 + 3-5 



1-3+5 0. 

Ilence, the quotient is a;^ — 3a; + 5. 
3. Divide 

afi + 2a^b + UW - a'^P - 2ab^ - Sb^ by a» + 2ah + Sb\ 
Ans. a3 + . a26 + . a62 _ 63 = a3 - R 



CHAP, XI. J SYNTHETICAL DIVISION. 887 

4. Divide 1 — a; by 1 -{- x. Ans. 1 — 2a; -h 2r» - 2*^+ &c 

6. Divide 1 by 1 — a;. Ans, I •{- x -\- x"^ + x^ -\- &c. 

G. Divide x'' —y'' by x — y. 

Ans. x^ -{- x^y -{- x*y^ + x^y^ + x^y^ + ^y® + y*. 

7. Divide a^ — 3a*a;2 + 'Sa^x* — x^ by a^ — Sa^a; + Sox^ — x^. 

Ans. a^ + Sa^x + Saa:^ + ^^• 

313. To transform an equation into another whose roots shall be 
the roots of the p-oposed equation, increased or diminished by a given 
quantity. 

A method of solving this problem has already been explained 
(Art. 264) ; but the process is tedious. We shall now explain 
a more simple method of finding the transformed equation. 

Let it be required to transform the equation 

ax"^ + Px"^^ + ^ar^-z .... Tx + U= 

into another whose roots shall be less than the roots of this 
equation by r. 

If we write y -\- r for a:, and develop, and arrange the terms 
with reference to y, we shall have 

aym 4_ p/ym-J _|. Q^ym-2 _ _ + ^/y 4. 17^ _ Q . . . (1), 

But since y = x — r^ equation (1), may take the form 

a{x-rY-\-P\x-rY-^ + Q\x - r)^'^ T{x -r)+ U'=(i (2), 

which, when developed, must be identical with the given equa- 
tion. For, since y -\- r was substituted for x in the proposed 
equation, and then x — r for y in the transformed equation, we 
must nece&sarily have returned to the given equation. Hence, 
we have 

a[x — rY + P'{x — r)'"-i + Q'{x — r)'»-2 . . . 7^ (a: — r) -f CP 
= ax-^ 4- Px"^^ H- §a;'^2 , , , Tx -\- U —0^ 

If now we divide the first member by x — r, the quotieni 
will be 

a(x - r)*"-! + P\x - r)'»-2 4. Q'^x _ ^j—a . _ ^, 
*nd the remainder JJ\ 



SS8 ELEMENTS OF ALGEBEA. ICHAP. XL 

Bui since the seo.ond member is identical \T.th the first, Iho 
very same quotient and the same remainder would arise, if the 
second member were divided by x — r : hence, 

J^ the first member of the given equation he divided by the unknown 
quantity minus the number which ex2:)resses the difference between th 
roots, the remainder will be the absolute term of the transformed equation. 

Again, if we divide the quotient thus obtained : viz., 
a{x — r)'»-i + F'{x — r)^"^ + Q'{x — r)"»-3 . . . ^ 
by X — r, the remainder will be T^, the co-efficient of- the term 
last but one of the transformed equation ; and a similar result 
would be obtained b}'- again dividing the resulting quotient 
by X — r. Hence, by successive divisions of the poly- 
nomial in the first member of the given equation and the quo- 
tients which result, by x — r, we shall obtain all the co-efficients 
of the traiisformed , equation, in an inverse order. 

Remark. — "When there is an absent term in the given equation, 
Its place must be supplied by a 0. 

EXAMPLES. 

Transfo'-^m the equation 

bx^ - 12a;3 4- 3a:2 -f 4a; — 5 = 
into anot? .?r whose roots shall each be less than those of the given 
equation ^y 2. 

First Operation, 
5x* - 122;3 -f 3a;2 4- 42: - 5 l | a; - 2 



5ar* - 102;3 5.^3 -2x^-x-^2 



- 2^:3 + 30:2 






- 2x^ + 4x^ 


-f 4ar 




- x^ 




- x^ 


-f 2a; 






2a?- 


-5 




2ar- 


-4 



— 1 1st remainder 



CHAP. XI. 1 SYNTHETICAL DIVISION. 389 

Second Operation, 
5a;^ — 2x^ — x + 2 I a; -2 
5x3 _ ioj;2 5.^2 -I- 8:c -h 15 



8^2- 


- X 




8x^- 


-IQx 






15^ + 


2 




Ibx- 


30 



32 2d remainder. 

Tliird Operation. Fourth Operation, 

5x^+ 8.C+15M x— 2 5a; +18 

5a:2_10a; I 5a; + 18 5a; - 10 



18a; 4- 15 28 4th remainder. 

18a; - 36 



51 3d remainder. 
Therefore, the transformed equation is 

5y* + 28?/3 -f 51?/2 -f 32?/ — 1 = 0. 
This laborious operation can be avoided by the synthetical 
method of division (Art. 312). 

Taking the same example, and recollecting that in the syn- 
thetical method, the first term of the divisor not being used, may 
be omitted, and that the first term of the quotient, by which 
the modified divisor is to be multiplied for the first term of the 
product, is always the first term of the dividend ; the whole of 
the worl: may be thus arranged : 

5-12 +3 +4 --5||2_ 



10 -4 



— 2 


- 1 


2 -1 


10 


IG 


30 


8 


15 


32 .-. T r= 32 


10 


86 




18 


51 


.-. C' = 51 


iO 






28 


,\P' 


= 28; 



??'=- 1 



890 ELEMENTS OF ALGEBRA. lCHAP. XL 

for it is plain that the first remainder will fall under the abso- 
lute term, the second under the term next to the left, and so 
on. Hence, the transformed equation is 

5y* + 2Sy3 + 5Iy2 -f 32y - 1 = 0. 
2. Fmd the equation whose roots are less by 1.7 than thos« 
of the equation 

a;3 _ 2a;2 -f 3a; - 4 = 0. 

First, find an equation whose roots are less by 1. 
1-2 +3 -4[|^ 
1-1 2 



- 1 2 -2 

1 



2 

1 

T 

We hare thus found the co-efficients of the terms of an equa- 
tion whose roots are less by 1 than those of the given equation: 
the equation is 

and now by finding a new equation whose roots are less than 
those of the last by .7, we shall have the required equation : thua, 

1 + 1 +2 -211.7 



.7 1.19 2.233 

r7 SA9 .233 

.7 1.G8 



2.4 4.87 

.7 



3.1 

hence, the required equation is 

2/3 -I- 3.1/- + 4.S7y + .233 = 0. 

This latter operation can be continued from the form.er, with- 
out arranging the co-efficients anew. The operations have been 
explained sepirata'y, merely to indicate the several steps in the 



CUAP. XL] SYNTHEPICAL DIVISION. 891 

transformation, and to point out the equations, at each step 
resulting from the successive diminution of the roots. Com- 
bining the two operations, we have the following arrangement: 



-2 


+ 3 


-4(1.7; 


or, 


1 -2 


+ 3 - 


- 4 (1.7 


I 


-1 


2 




1.7 


- .51 

2.49 


4.233 


-1 


2 


-2 


- .3 


.233 


1 





2.233 




1.7 
1.4 


2.38 

4.87 







2 


.233 




1 


1.19 






1.7 
3.1 






1.7 


3.19 




.7 


1.68 













2.4 4.87 
.7 

ST 

We see, by comparison, that the above results are the same 
as those obtained by the preceding operations. 

3. Find the equation whose roots shall be less by 1 than 
the roots of 

a;3 - 7a; + 7 = 0. 

Ans. 2/3 -f 3y2 _ 4y _f- i — Q^ 

4. Find the equation whose roots shall be less by 3 than 
the roots of the equation 

x^ — 3x3 __ 15^2 _|_ 49^ _ 12 =, 0. 

Ans, ?/* + 9y3 -I- 12^2 — 14?/ = 0. 

5. Find the equation whose roots shall be less by 10 than 
the roots of the equation 

X* + 2.r3 _^ 3^2 + 4a; — 12340 = 0. 

Ans. 2/4 + 422/3 -l G632/2 + 4664y = 0. 

6. Find the equation whose roots shall be less by 2 than 
the roots of the equation 

a;5 ^ 2.^3 — Ga:2 _ 10^ _ 0. 
Ans. 7/5 ;. 102/* + 4-22/-- -f 862/^ -j- 70y -f 1 = 0. 



392 'elements of algebra. [chap, xi 

Horncr''s Method of approxrnating to the Real Roots of 
Numerical Equa/Aons. 

314. The method of approximating to the roots of a nnmerl 
cal eauation of any degree, discovered by thp English laathe. 
matician W. G. Horner, Esq., of Bath, is a process of very 
remarkable simplicity and elegance. 

The process consists, simply, in a succession of transform* 
tions of one equation to another, each tcansformed equation, as 
it arises, having its roots equal to the difference between 
the true value of the roots of the given equation, and 
the part of the root expressed by the figures already 
found. Such figures of the root are called the initiul fgures. 
Let 

F=ar'»-l-P.r'»-i-f- ^^^"-2 . . . . + ra; + Z7=0 - - - (1) 

be any equation, and let us suppose that we have fouu-* a 
part of one of the roots, which we will denote by m, and d^ 
Qote the remaining part of the root by r. 

Let us now transform the given equation into another, wLose 
roots shall be less by m^ and we have (Art. 313), 

V = r^ + R'r^'^ + $V^2 .... -I- T'r -\- U' = ■ (2). 

Now, when r is a very small fiaction, all the terms of tho 
second member, except the last two, may be neglected, and tho 
first figure, in the value of r, may be, found from the equaiiou 

U' IT 

T'r + f/' ^ ; giving - r = — ; or r =r - •- ; hence, 

The fir 8t figure of r is the first figure of the quotient obtained by 
dividing the absolute term, of the transformed equation by the penulti' 
mate co-efficient. 

If, now, we transform equation (2) into another, whose roots 
shall be less than those of the previous equation by the first 
figure of r, and designate the remaining part by 5, we shall 
have, 



CHAP. XI. I nORNER's METHOD. 393 

the roots of. which will be less than those of the given equa- 
tion by m + the first figure of r. The first figure in the valuo 
of s is found from the equation, 

r^s + U^^=:0, givbg s=^. 

We may thus continue the transformations at pleasure, and 
each one will evolve a new figure of the root. Hence, to find 
the roots of numerical equations. 

1. Find the number and places of the real roots by Sturm-s* 
theorem^ and set the negative roots aside. 

[I. Transform the given equation into another whose roots shall 
he less than those of the given equation, bjj the initial figure or 
figures already found: then, by Sturms' theorem^ find the places 
of the roots of this new equation, and the first figure of each will 
be the first decimal place in each of the required roots. 

111. Transform the equation again so that the roots shall be lens 
than those of the given equation^ and divide the absolute term of 
the transformed equation by the pemdtimate coefiicient^ which is 
called the trial divisor^ and the first figure of the quotient will be 
the next figure of the root. 

IV. Transform the last equation into another whose roots sUall 
he less than those of the previous equation by the figure last found., 
and pi'occed in a similar manner until the root be found to the 
required deg7-ee of accuracy. 

Remark I.— This method is one of approximation, and it may 
happen that the rejection of the terms preceding the penultimate 
term will aficct the quotient figure of the root, To avoid this 
source of error, find the first decimal places of che root, also, 
by the theorem of Sturm, as in example 4, page 399, and when 
the results coincide for two consecutive places of decimals, those 
Bubsequently obtained by the divisors may be relied on. 

Remark II. — When ^he decimal portion of a negative root is 
to be found, first transfoi'm the given equation into a'.other by 
changing the signs of the alternate terms (Ar'.. 280), and then 
find the decimal part of the corresponding positive root of 
this new equation. 



894 



ELEMENTS OF ALGEBRA. 



[CHAP. XI. 



+ + no variation, 
— + three variations; 



III. When several decimal places are found in the root, the 
operation may be shortened according to the method of com 
tractions indicated in the examples. 

314i Let us now work one example in full. Let us take the 
equation of the third degree, 

2,3 _ 7^ + 7 ^ 0. 
By Sturm's rule, we have the functions (Art. 299), 

A'l = 3x-2 - 7 
Xc = 2a; -3 
X3 = + 1. 
Hence, for ar = 00, we have + + 

iC = — OD " — + 

therefore, the equation has three real roots, two positive and on© 
negative. 

To determine the initial figures of these roots, we have 

for a; = ... H h for a: = . . . H f- 

ar=l... + + a;=-l... + + 

a; = 2...-f + 4-+ a;=-2. ..+ + - + 

a::=-3... 4- + - + 
a:=-4... --{- - + 

hence there are two roots between 1 and 2, and one between 
— 3 and — 4. 

In order to ascertain the first figures 
in the decimal parts of the two roots 
situated between 1 and 2, we shall trans- 
form the preceding functions into others, 
In which the value of x is diminished by 
1. Thus, for the function X, we have 
this operation: 

y = ifi •\- 3?/2 4?/ -J- 1 • 

And transforming the others in n \ j j v % 

ine same wav we obtain the • ^ ^ » 

A .• 1^2 = 2?/ -1; 

fUncticns -^ ' 

1^3 =+ 1. 



1 + - 7 + 7 (1 
1 + 1-6 


1-0 + 1 

1+2 

2-4 

1 

3 



CHAP. XI.] 



HORNER S METHOD. 



895 



Let y = .1 "vre have H f- 



y = .2 
y = .3 

y = .4 
y = .5 
y = .6 

y = n 



two variations, 



one variation, 



+ -- + 
+ -- + 
. + 

=F + 

- + 4- + 

+ -f -7- -{- no variation. 

'Ilierefore the initial figures of the two positive roots are 1.3, l.(J, 
1 et us now find the decimal part of the first root. 



hO 

1 

T 

1 

2" 

1 



*3.3 
3 

"3^6 
3 



**3.9 5 
5 

4.0 
5 

***4.0 5G 

C 

4.0 62 

6 

•♦»*4.01G8 8 



|4.0|69e 



-7 
1 

2 

*-4 



.99 



-3.01 

1.08 

**-1.9 3 

.1975 

-1.7325 

.2000 

***— 1.5325 

.024336 

-1.508164 

.024372 

****_ 1.4837912 

.OO325I4 



1.48053 
.00325 



— 1 .4772 
.0003 



— 1 .4769J2 
000316 

^1| 4|4|7|6]5 



+ 7 (1.356895867 

-6 

~*T 

-.903 



**,097 

— .086625 
*«* .010375 

— .009048984 

**«* .001326016 

— .001184430 

.000141586 

— .000132923 

.000008663 

— .000007382 

.000001281 

-.000001181 

.000000100 

— .000000089 

.000000011 
-.000000010 

i 



396 ELEMENTS OF ALGEBRA. [CHAP. XJ 

The operations in the example are performed as follows: 

1st. We find the places and the initial figures of the posi- 
tive roots, to include the first decimal place by Sturms' theorem. 

2d. Then to find the decimal part of the first positive root^ 
we arrange the co-efficients, and perform a succession of trans- 
formations by Synthetical Division, which must begin with the 
initial figures already known. 

We first transform the given equation into another whose 
roots sliall be less by 1. The co-efficients of this new equation 
are, 1, 3, —4 and 1, and are all, except the first, marked by 
a star. The root of this transformed equation, corresponding 
to the root sought of the given equation, is a decimal frao 
tion of which we know the first figure 3. 

We next transform the last equation into another whose 
roots are less by three-tenths, and the co-efficients of the new 
equation are each marked by two stars. 

The process here changes, and we find the next figure of 
the root by dividing the absolute term .097 by the penulti- 
mate co-efficient — 1.93, giving .05 for the next figure of the root. 

We again transform the equation into another whose roots 
shall be less by .05, and the co- efficients of the new equation 
are marked by three stars. 

We then divide the absolute term, .010375 by the penultimate 
co-efficient, — 1.5325, and obtain .006, the next figure of the 
root : and so on for other figures. 

In regard to the contractions, we may observe that, having 
decided on the number of decimal places to which the figures 
in the root are to be carried, we need not take notice of 
figures which fall to the right of that number in any of the 
dividends. In the example under consideration, we propose to 
carry the operations to the 9th decimal place of the root; 
hence, we may reject all the decimal places of the dividends 
after the 9th. 

Tlie fourth dividend, marked by four stars, contains nine 
decimal places, and the next dividend is to contain no morew 



CHAP. XL] Horner's method. 397 

But the corresponding quotient figure 8, is the fourth figure 
from the decimal point ; hence, at this stage of the operation, all 
the places of the divisor, after the 5th, may be omitted, since 
the 5th, multiplied by the 4th, will give the 9th order of deci- 
mals. Again : since each new figure of the root is removed • 
one place to the right, one additional figure, in each subsequent 
divisor, may be omitted. The contractions, therefore, begin by 
BtrikinfT off the 2 in the 4th divisor. 

in passing from the first column to the second, in the next 
operation, we multiply by .0008 ; but since the product is to 
be limited to five decimal places, we need take notice of but 
one decimal place in the first column ; that is, in the first 
operation of contraction, we strike off, in the first column, the 
two figures 68 : and, generally, for each figure omitted in the 
second column, ive omit tiuo in the first. 

It should be observed, that when places are omitted in either 
colunm, whatever would have been carried to the last figure 
retained, had no figures been omitted, is always to be added 
to that figure. Having found the figure 8 of the root, we need 
not annex it in the first column, nor need w£ annex any sub- 
sequent figures of the root, since they would all fall at the 
right, among the rejected figures. Hence, neither 8, nor any 
subsequent figures of the root, will change the available part 
of the first column. 

In the next operation, we divide .000141586 by 1.4772, omit- 
ting the figure 8 of the divisor : this gives the figure 9 of the 
root. We then strike off the figures 4.0, in the first column, 
and multiplying by .00009, we form the next divisor in the 
second column, — 1.4769, and the next dividend in the 3d 
column, .000008663. Striking off 5 in this divisor, we find 
the next figure of the root, which is 5. 

It is now evident that the products from the first column, 
will fall in the second, among the rejected figures at the right ; 
we need, therefore, in future, take no notice of them. 

Omitting tha right hand figure, the next divisor will be 1.476, 
and the next figure of the root 8. Then omitting 6 in the 



898 



ELEMENTS OF ALGEBRA. 



LCHAP. XL 



divisor, we obtain the quotient figure 8 : omitting 7 we obtaiD 
6, and omitting 4 we obtain 7, the last fig are to be found. We 
have thus found the root x = 1.356895876 . . . . ; and all simflai 
examples are wrought after the same manner. 

Tlie next operation is to find the root whose initial figures are 
].6, to nine decimal places. The operations are entirely similar 
to those just explained. 

We find for the second root, x = 1.69202141. 

For the negative root, change the signs of the second and 
fourth terms (Art. 280), and* we have, 



1 


-0 


- 7 




— 7 (3.0489173396 




3 


9 




+ 6 




3 


2 




~ 1 




3 


18 
20 




.814464 




— .185536 




3 


, 


3616 


.166382592 



9.0 4 


4 

9.0 8 


4 


9.128 


8 


9.136 


8 



20.3616 
.3632 

20.7248 

73024 

20.797824 

73088 

20.8709112 

82310 



— 19153408 
18791228 

-362180 

208875 

— 153305 
146212 



..|9.1|44 



20. $7914 

823 



20.8873 



20.8874,6 
I9 



2|0.!S|8|715 
4. Find the roots of the equation 

a?3-}- Ux^'-I02x+ 181 =0. 



-7093 
0266 

-827 
626 

-201 

188 

12 



« 



CHAP. XL] HORNER'S METHOD. 399 

X 

The functions are 

X = x^ + 11.^2 _ I02:c + 161 

Xi = 3^2 + 22^ - 102 
X, = 122:c - 393 
X, = + ; 
and the signs of thfi loading terms are all + ; hence, the sub- 
■titiildcn cf — X and -f- '^ must give three real roots. 

To discover the situation of the roots, we make the substitu- 
tions 

a? z= which gives -\ h two variations 

x=l " H h " 

x = 2 « H h " 

x = Z « 4- -j- « 

X = 4 " +-}- + + no variation ; 

hence the two positive roots are between 3 and 4, and we must 
therefore transform the several functions into others, in which J 
shall be diminished by 3. Thus we have (Art. 314), 
r = 2/' H- 20y2 - 9y + 1 
r; = 3y2 -f 40y — 9 
Y, =z 122y - 27 

Make the following substitutions in these functions, \iz. : 
y = signs -\ f- two variations 

y = .l " + + 

y = .2 " 4- + « 

y =z .Z " + + + + no variation ; 
hence, the two positive roots are between 3.2 and 3.3, and wf 
must again transform- the last functions into others, in which y 
ghall be diminished by .2. EfTectiug this transformation, we have 
Z = z^ -{- 20. Qz^ - .SSz -f .008 
Zi = 3^2 + 41.22 - .88 
Z, = 1222 - 2.6 
Z,= -f. 



400 



i:LEMJ:N-TS "^OF /ALGEBRA; ^ 



[CHAP. XL 



Let z = then signs are -\ + two variations, 

z = M " " -i h " 

g = .02 " " h one variation, 

= .03 " " -h 4- + + no variation ; 

nence we have 3.21 and 3.22 for the positive roots, and the sum 
of the roots is — 11 ; therefore, — 11 — 3.21 — 3.22 = — 17.4, 
is the negative root, nearly. 

For the positive root, whose initial figures are 3.21, we have 
X = 3.21312775 ; 
and for the root whose initial figures are 3.22, we have 

X = 3.229522121 ; 
end for the negative root, 

x= - 17.44264896. 



EXAMPLES. 



1. Find a root of the equation a;^ _^ 3.2 ^ a. _ jqq _ q^ 

Ans. 4.2044299731. 

2. Find the roots of the equation x^ — 12a:2 + 12.r — 3 = 0. 

+ 2.8580S330S163 
-I- .606018300917 
+ .443276939605 

L - 3.907378554685. 

3. Find the roots of the equation x*— Sx^ + 14.1"^ 4-4a; — 8=i0l 

+ 5.2360079775 



Ans. 



Ans. ^ 



+ .7639320225 
+ 2.732050S075 
- .7320508075. 



% Find the roots of the equation 

a;5— 10a;3 + 6x+ 1 



0. 



Ans, 







559 



iP 



^ - 3.0653157912983 

- ".6915762804900 

- .1756747992883 
-f- .8795087084144 
-f 3.0530581626622. 



wmmu 




.:.MmM 



m 









,ri 










at iiiiil 

Mm 



